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How to deal with a very weak student?
I am teaching an honors class for seniors (at a very good university, top 5 in my field). Most students are fantastic. They are very interested, they often surprise me with insights in class, and the class is very lively not because I am a particularly engaging lecturer but because the students love what they are learning. They thrive on painful homework sets and difficult homeworks/exams. They can't get enough of whatever I throw in their way. In all, they are a joy to teach.
Unfortunately, in a class of such strong students, the weak student sticks out like a sore thumb. One student in particular is doing extremely poorly. Generally homework sets are well done since students more or less help each other out, but in quizzes and exams this student has scored failing grades (10-20% range). But because of her good performance in the homework sets, I don't think that she realizes that she is a very weak student.
This in itself is fine, because not everyone can be strong in everything. But what worries me is the subconscious that manifests in her behavior.
For example, she comes to my office hour (she is the only one that comes regularly; I hold three hours' office hour a week, and she is there all three hours. The others come and stay for no longer than 10-15 minutes).

She tells me that a particular problem is not doable because it was not covered in class (it was done and I point it out in her notes that she took in class). Her favorite complaint is that a problem seems unapproachable unless we know some advanced topic (that she gleaned from the textbook) that will not be covered for a few more weeks. I ask her to prove it using the advanced topic and she still cannot do it; furthermore, the problems are most definitely doable with what I have covered in class.
When I do not post the solutions immediately after the homework sets are due (because as a professor, I happen to have things other than teach this course), I expect an email in my inbox asking where the solution set is.
When she does not understand a concept, she bring her notes and says that the class was unclear and that I should explain it again to her (I receive great evaluations, and I am in particular noted for the clarity of my lectures).

The thing that I am concerned about is that less than a month into the semester, she is already showing signs of blaming her problems on me. I know her very well academically, since we have been spending three hours a week going over the course material. There is no chance that she will pass this course (as she has no idea how to think for herself; her idea of academic improvement is to consistently show up to my office hour and listen to me talk), and I am worried about her creating problems for me. She is a transfer student from a community college, and no one else has any data on her as this is her first semester. Furthermore, as a woman, I often find that students tend to demand things from me, and I am seeing signs of this (asking for more office hours in a completely entitled way, etc.)
It would honestly improve the quality of my life so much if I could somehow find a way to deal with her. What can I do to 1) try to get her to drop my course, and 2) cover my ass for the administrative hell she might raise, if she fails my course?
As a disclaimer, I have had weak students before, and I was not this worried about them, because they acknowledged that they were weak, and they actually put work into trying to learn the course. This one, however, seems more into thinking about what things are in her way of learning that I am not doing.
EDIT: Thank you for your sympathy and many insights to how I might deal with this issue. It took me a long time to read everything, but I have decided to do the following:

Take a harder line of approach with how she interacts with me in office hours; I will guide her to ask very specific questions, instead of the generic "I didn't get this part of the lecture" and if she protests, I will voice my concerns over her poor performance in quizzes. But I do think that she is not doing anything wrong by taking up all my time. I just have to be more firm in choosing which questions to answer and which not to answer.
Document her questions and my answers/approaches to her questions.
But since she has been scoring nearly perfect scores in her homework sets (although she is failing her quizzes badly), I do not think that I will suggest that she drops my course for now. The add/drop deadline passed about a week ago, so there is no alternative for her anyway, and I will suggest that she drops the course after the midterm. Or maybe some miracle will happen and she will actually do well!

@user37208 Yes, there is. But is suggesting that option kosher? Since I am not planning on having a major evaluation until mid-to-late October, I think that it might be offensive to suggest something like that without having solid evidence (the rest are small pop quizzes/foretold quizzes which she did extremely poorly on, and the homework sets, which she has done decently on).

So many answers that apparently do not understand what an honors course is and answers as if this was some random undergrad course which was required for all students...

I may be way off base here but ... You don't mention in what country you are, or what the student's cultural background might be, if different than yours or the other students. This may not be a factor, but can sometimes be an enormous one.

I once had a student who didn't really have the background for the course she was taking, but she worked hard, came to almost every office hour with real questions (i.e., not "how do you do this problem?" but "here's how I tried to do this problem and here's where I got stuck"), and finally got an A in the course.  This is one of my happiest teaching memories.

@jpmc26: Of course! But everybody might be better off if she failed maths 101 instead of a very teaching-intensive honors course. For some courses it might also be better if students are kicked out after failing the first (sub-)exam, or at least seeing they're scoring 15%.

The question is, how could she enter an honor course? Maybe she was accidently placed there because of delivering good homework? Then the mistake was probably made before. I cant answer your questions as I am or rather was a student myself. But I pull off my hat for your efforts and thoughts. You seem to be one of the good guys.

@VonBeche The problem is that she could probably pass Math 101 with this approach. It is primarily in the more advanced courses where it starts to break down.

I'm going to point out the obvious but somewhat cruel thing: she is cheating on her homework.  She clearly lacks the chops to do it herself, so someone else is doing it for her.  Cheating is probably ground for removal from the university, let alone just your class.  If you want to be cold-hearted about it, confront her with a problem similar to one she solved in a homework set; when she is unable to do it, offer her the chance to drop the course to avoid an investigation.

@Malvolio Unfortunately, in theoretical courses, cheating tends to be fairly rampant among all students. The university administration tends to turn a blind eye towards it too, because 1. it would affect most students, and 2. unless there are concrete evidence, it is very hard to prove that the student was cheating. So instead, many people make the homework worth very little of your grade, 10-15%, and the rest are tests.

@Sana -- that's what I mean about "cold-hearted".  Yes, she isn't doing anything anyone else isn't doing, but you can catch her at it; yes, the administration might want to ignore it, but she doesn't know that.  Shown obvious proof of her own misdeeds, and offered an easy way out (dropping your class), she'll almost certainly take it.

Bring it up in class, that you don't disapprove of cooperation on homework, but that those taking part in it assume a responsibility that everyone who has a solution has also understood it. Letting others just copy is actually self-righteous foul play.

Dunning-Kruger would suggest that she doesn’t know that she can’t do it, and polite hits won’t change that.

Some short-ish advice:

First: Realize that you can't save everyone. I'm on the other end of this, teaching at the community college, which is something like an academic M.A.S.H. unit. Some people you have to realize are, most unfortunately, unsaveable ("black-flagged" in medic parlance). The weakest students are, oddly, the ones most resistant to suggestions about switching to easier courses (I tried this twice this semester to no avail: see Dunning-Kruger effect).
Second: Set boundaries. It's probably an abuse of your office hours that this one student is taking up all of them. Proper usage is, as you say, 10-15 min for a particular question or issue. For a particularly weak student like this I may say, "Let's say we have 30 minutes for this." Require that she ask about a particular homework exercise that she can show prior work for. Do not just regurgitate the lecture wholesale.
Third: One student failing should not cause administrative hell. Students will fail and this is a normal part of the institution's function. If she does complain then (at least where I am) the instructor is given the benefit of the doubt. The majority of the time, a student in this situation has a record of failing other STEM courses. (If it does cause hell, then you have my enormous condolences for a degenerate administrative system.)
Fourth: It doesn't hurt to document what you efficiently can. I keep a digital gradebook via the learning management system; occasionally a student complains, my department ask for documentation, I turn that over (showing multiple documented failures), and that's the last I hear of it. After grading, I actually run all my tests through a bulk digital scanner so I keep a copy before I hand them back (in fact: I make all my tests on one sheet of paper to assist this; takes about 1 minute at the school's feed scanner). This actually hasn't been seen by anyone else in the few years I've been doing it, but it's there if I do need it. Also it's good for me to reference as documentation and examples in the future.

Thanks for your advice. A follow-up question: If I am setting aside three hours a week for office hours anyway, is it OK to suggest to the student that I am not willing to spend all of those times with her, even when there is no one else needing my attention? Most other students are very strong and only very few students come to my office hours. So she is technically not doing anything wrong by taking up all my time.

@Sana: I would say yes. In fact, I would say that she definitely is doing something wrong (and equivalently: that you should change your thinking about that as a boundary). If you find it easier to communicate, set a very narrow frame for what's acceptable: questions about specific exercises she can show work on. If she can't show that, tell her to go work on homework and if she has a specific blocking point come back. Do not accept "I can't even get started"; the response to that is, "Look at the example in your notes".

@DanielRCollins Already tried that, unfortunately. She then comes back with, "I didn't understand what you did in class. Can you go over this part of the note again?" That aside, I do understand what you mean by boundary, and I will definitely have to think hard about this before my next office hour.

You could suggest: "We have 30 minutes today - please select the questions you want discussed in this slot." Give her the choice. If she asks why 30 minutes, you say, it would be otherwise unfair to the other students (which it actually is - if she were to succeed, it is because she has you as very competent "group member").

Look at it this way: students should in principle be able to understand the material based on lectures and homework without office hours - that's the implicit default. If she consistently takes up hours of your office hours, there is an obvious mismatch between her ability or her command of prerequisites. If the other students don't struggle, then the problem can't be with the content or the instructor. Perhaps arguments like this could be helpful in discussing dropping the course with her.

@Sana Your question insinuates that her homework is done with other classmates.  Why can't she ask them for additional explanation when her notes are insufficient?  Sometimes other students can do a better job of "making it click" than the prof.

This is a good answer.  I especially second the part about documenting everything.  I would even document your concerns outside of your own records, possibly in an email to the student or to academic support maybe; whatever seems more appropriate.  If you have reason to fear backlash, things like that to point to should make your life much easier.

One technique I use to let a student know that office hours are over for them, and they should please leave, is to congratulate ourselves on the roadblock we have removed.  "So, good! Now we understand that blah, blah, and blah.  Why don't you go and hack away at this some more and see what else you can clear up."  I can imagine that some students will not take that hint, although I have gotten pretty successful with it.  For students that are still monopolizing time, a frank discussion about grade expectations given current difficulties could be in order.

@Sana: I disagree with the other answers; if there are no other students waiting, I think it is entirely fine for one of my students to take up as much of my office hours as they want. In fact, in an honors course, I strongly encourage them to!

Another way of looking at office hours - she's getting three extra class hours per week of instruction. Is that fair to the other students who are only receiving the other classroom instruction time?

@TomChurch I am in agreement with you, and I am shocked that most people think that I am entitled to kicking her out after a certain amount of hours, since that is what the office hours are for! However, I think I would be more careful in making sure that my office hour does not become a re-iteration of my lecture, and I think I will make sure that the student is talking for the most part, and that my role in office hours is just to dispense small hints.

This is all good advice because the worry I see here is that when she fails and complains, she will have a mountain of evidence about how she approached you all the time and put a TON of effort into fixing her problems yet still failed and thus it's clearly not her fault. I would not underestimate the power of restricting contact and document. It should help you sleep easier if you don't have to deal with her all the time too

Assuming there is some progression in the homework problems, next time you get "I don't know how to approach this," pull out a copy of the precursor problem, on which she presumably got good marks.  Say, "The problem you ask about follows from this one.  Explain to me, without referring to what you turned in, how you approached it."  If she can, then maybe she can be saved.  If she cannot, she'll have to realize that she cannot, and that you know it.

@SLC yes this is super helpful and I have begun documenting our interactions.

@Sana The issue isn't that she's the only one there, but why she is the only one there.  Sometimes, her always being there is the reason she's the only one there (most of the time).  Some students see "that student" is in the office and decide they'd rather not deal with that and abandon asking their questions.  They just might not like her personality; or they get a vibe while she's there that makes them uncomfortable; or they decide "yeah, she really needs it, and I don't need it nearly as badly, so I can go without", thus depriving themselves; etc.  You have a duty to stop that.

Even moderately decent students can identify a student that just cannot possibly get it just as well as you can, and they tend to be put off when such students are granted a monopoly on your time and help.  I would personally worry a lot less about how this one student's opinions are going to affect how the department views your teaching, and instead worry about how all of your other students' opinions of your interactions with this one student is going to affect and color their opinions on your teaching.  That stands out a lot more on reviews.

@SohaFarhinPine Please do not impose your personal style to others

I haven't seen this point mentioned yet:

She is a transfer student from a community college, and no one else has any data on her as this is her first semester
When she does not understand a concept, she bring her notes and says that the class was unclear and that I should explain it again to her
her idea of academic improvement is to consistently show up to my office hour and listen to me talk

She is doing things that work well in secondary school (a.k.a. high school) - she's working hard on her homework and making maximum use of your office hour, and so on. She probably thinks she's working hard and doing well. In school, the exam questions tend to test whether you've learned exactly what was told, not more.
But at some point a student has to learn that university isn't secondary school. It's much more about working on your own than about absorbing from a teacher. Not everybody knows that when they start. She doesn't realize she needs to change her way of studying.
So I think you could also have a conversation on that, she's there in your office anyway.

In line with your comment that university isn't secondary school, I found this video very interesting, in particular the concept of metacognition.  Samford University apparently made a set of these videos to facilitate new student's transition to university learning and expectations.

The combination of these three is a class of students which is very familiar to me. Because of some issues (unknown to me, but perhaps mental), these students are not able to focus and work hard on understanding the concepts and try to rely heavily on others to clarify everything.

I see your point, but where do high school teachers have office hours? Maybe "lower division course" would be a better description.

Not a full fledged answer, but some tips:

Prepare appropriate responses to complaints. Do not defend yourself, but state your expectations. For example, for "This question is not doable." do not answer "But is doable like this..." but respond "Being able to do questions like this is part of the expectation for the course." For "I did not understand the class and I have no idea where to start." answer "Students in this class are expected to work through the material and be able to grasp the course material. The office hours are for additional questions and occasional clarification."
You may take some time to review her performance with her based on facts. Something like "Let's see if your difficulties with the questions are reflected in your other performance. On previous homework you did like...  while in the quizzes the picture is very different. Note that the final grade will be determined like...  and that your current performance may not be enough. To be able to pass this course you need better performance in... " For the last point it is important to not judge the person but her performance.
As for "I have no idea where to start." Use the Polya questions "What is the given data?" and "What would the answer be?" Do not answer these questions under any circumstances but stress that every student has to be able to answer these questions on his own. You may help a bit and reiterate the question several times (from my experience, most weak students need to hear the question more than two times before they even start to think about the answer) but you shall not provide the answers even if office time is over and not even if it is the second or third time the student comes to office hours with this question.
Ignore the remarks like "The lecture was unclear." and respond "What question do you have about the class." Focus solely on the student’s learning.  Sometimes I would go as far and say "If you don't have questions about the class then I can't give answers."
Do not suggest an easier class but try to make it her idea. For example you could drop the line "The expectations for this course is...  but that course has the expectations..." Be sure to not imply that she will make that other course for sure.

Is it currently mathematically impossible for her to actually pass the course? Or are you just assuming that she won't based on her current work? Either way, you do have a responsibility to let her know. If I were her and I were failing and knew there was nothing I could do, I'd be out of your hair faster than you could blink.
As far as your office hours, I don't think it's particularly good to change them this semester just because you have a student that's abusing them. In the future though, you may want to consider stating that your office hours are a 3 hour block of time during which a student can reserve 30 minutes.
It sounds to me like she's not really paying attention in class at all. Have you put her in the front of the room?
I've had a few students before who want to show up to office hours with the expectation that you'll just do their work for them. As far as her excuses and table-turning behavior, you should address it in a way that is very matter-of-fact without emotionally frontloading it. "Yes, we did this in class." "If I tell you that you won't learn how to do it." Do. Not. Acquiesce. If she continues to struggle, have her break it down into the smallest possible components and go from there. It's okay if they don't like you. It's not okay if they take advantage of you and you allow it.
I agree that you need to be documenting her time with you, what you do with her when she's there, and her progress. For your own protection.
Perhaps you should suggest to her that she gets a tutor that isn't you? If you have a recommendation ready to go, maybe she'll just take that and go work with a tutor.

Hope that helps - just remember that at the end of the day you can be responsible for what you offer to someone's education, but you can't be responsible for their talent, work ethic, or intellect.

To add to point #2, it's hardly ever a good idea to introduce an overall rule for everyone because of the bad behavior of one or two people. It's better to confront the one or two people and not allow them to abuse the system. An accretion of ad-hoc rules leads to a complex and fragile system.

Daniel's answer is good, but here are some different suggestions.

Contact the student's academic advisor (maybe there is a special one for transfer students) as well as the honors college to make them aware of the issue.  (It sounds like this student should not be in an honors course.)  They may have suggestions and be able to talk to the student.  (Also this will help in the event of your concern 2, but I suspect it's an unlikely concern.)
Ask your department head for advice.  (Again this should also help protect you in the event of 2.)
Consider having a conversation with the student.  I sometimes do, and it may or may not have any effect.  (Based on your description of the student, I would guess no effect except maybe making her upset, but that doesn't mean she shouldn't be confronted with these issues.)  I wouldn't tell directly the student to drop or that they will definitely fail, but I'd point out a couple of concerns (exam/quiz scores, perhaps a lack of expected preparation) and note that the amount of time the student spending on this course may not give them enough time to do well in their other courses, and suggest the student think about it seriously.  (Ideally this would happen at the beginning of the semester.)

I think this answer deserves some extra attention, especially the first two points. I know this isn't the case in all institutions, but at mine I would absolutely expect to hear if one of my advisees was having this kind of problem. I would also expect our Early Warning system to be invoked. The OP should find out what kind of support their institution offers for students and faculty, and take advantage of it--we don't have to do it all on our own!

I can totally sympathise with you and the stress you must be going through right now.
A few suggestions that might help:

I would suggest that you always see this student with another person (academic, etc) present as a witness to all your efforts and your attempts at helping her. I know you document your discussions, but it might help if somebody else was present too as a witness.
Why not find some really good and self motivated students in your class and pair her up with them? If they can't sort out her queries, then try a post-graduate student instead. If all else fails, then she can come to you. I always find poor students often do better when working in the company of other good students.
As already mentioned, explain to her that you're concerned about her future performance on the course and she might want to consider changing to another course.

Sounds like you're already doing a lot to help her-unfortunately, you can't change every student.

(1) means wasting not only your own but other peoples time on that one student. That's like keeping the door open when talking to a female student (as a male prof). It is actually insulting on that student. If you have so little faith in justice and your administration, i recommend a therapist, or relocation.

@Karl It's an extreme measure perhaps in the present case, but saying that it is an insult? There are institutions which require you to keep the door open when you talk to a student.

@CaptainEmacs By getting in a second person, or keeping the door open, you are insinuating that the student would otherwise raise false accusations on you. That is either a personal insult, or you are at a place with an institutionalised "we can't trust anyone" climate. No?

@Karl There are institutions (esp. in the US) which have such a policy. It's not the individual's decision. Some of these are high ranked and while, yes, I find that uncomfortable, many things that one used to be able to treat in the framework of "good/bad manners" have now been elevated to actionable contraventions. Not my personal preference, but that's how it is. In the present case, it is not clear whether the student is as untrustworthy as the response might insinuate. However, even the best of us may encounter cases which they'd rather meet with witnesses. I agree: only as a worst case.

@CaptainEmacs Well, my analogy to that bigot "open door" policy is a weak one, actually. While accusations of sexual misconduct always leave a stain, an upset student who failed a course is not likely to impress anyone. I guess.

@Karl What bigotry means depends on times. My personal term for this type of view is "Neopuritanism" or "Neovictorianism". Our society look down at past social norms as bigoted which then were perfectly acceptable and for whom our attitude is completely out of order. To stay within the current topic: elitism and "selection of the best" was perfectly acceptable, say, 25 years ago, and now it is considered unfair and discriminatory for disadvantaged students (no matter whether the disadvantage is gender, background, and, yes, personal talent).

The situation discussed here is precisely a consequence of this shift in what is considered the socially acceptable norm. When I studied, a weak student with "entitlement syndrome" would have been treated harshly and with no or very little effort to address their issue. Today, this is considered unacceptable. This shift in power towards the students causes reactions in form of "open-door policies" (of course, in parallel as to how harassment by authority persons is now being seen differently). Calling it "bigoted" is, in my opinion, a too simple description of a complex balance-realignment.

@CaptainEmacs Agreed. I called it bigotry, because in my opinion, we switched from "not care at all" to just "make it look so that nobody can accuse us of not caring". In the given example, nobody cared to make sure the student was in a course he's prepared for,

@Karl: Indeed - "making sure to be seen to be doing the right thing" is probably the most generic description of hypocrisy, independent of prevailing social code. In times of socialism (i.e. financial communism - everyone expected to be equal on money and compete on intellect/achievements), scientists in humanities (but also in STEM) were pressed to support dialectic materialism (I still own books attesting to that). Today, intellectual communism prevails (where everyone is expected to be equal on intellectual achievements, and compete on money), we are pressed to support conformist idealism.

Suggesting that she could pair up with one or more of the stronger students is a good strategy, they will be the first to suggest that the course is beyond her capabilities. Sometimes  a fellow student will understand where the sticking point is, better than a teacher, and show the solution. Third, the student in question might realise what a great teacher you are if her peers' explanation do not nothing to lift the fog.

I had a similar experience and I behaved like the given answer of @Daniel. The important fact that we should obey point about our teaching styles. In my class, the average student asked me to show more applied examples via the software in the class that I avoided. Also,  "I didn't understand what you did in class" is a familiar expression for the average students.
In the next term, I've changed my style of teaching including more examples with full details, giving more time and skip some material from the contents and provide in-time solutions for Homeworks. The result was surprising for me due to the excellent evaluation of my class which was the best among 6 years of teaching experience.
Just give more time on your explanation the ideas in the class with more applied examples (including computer-aided examples, simulation, and playing real stories in the class and etc!) even if you have to sacrifice some material from the course topics.

From what the OP writes, this sounds like making the class less interesting to 95% of the students in order to make it more accessible to a single student. All this in the context of an Honors class...

"Less interesting" is a weak verdict. Substancial lowering of standards I would call it. Evaluation by current students is widely regarded as irrelevant, btw. You have to ask former students, if your course prepared them for future task, not the current crew, which will give you points for funny stories and making life easy for them!

I just wanted to offer an alternate perspective on this thread from personal experience.  I think there's been a lot of great advice on this so far and I agree 100% that if the student has an entitlement problem and is making you misersable then yes, you should take steps.
That being said, your question made me feel sad because when I was in graduate school, I might have had a professor that considered me to be your weak student.  It was my first semester and I was taking the class to fulfill a Masters/PhD requirement.  I had come back to school after working professionally for nearly a decade, so I was rusty on a number of topics, but I had decided that if I just worked hard enough then it wouldn't make a difference.
I really struggled with one of my classes in the first semester.  I was there for every single class, every single office hour of the professor, and every single office hour of the TA.  I spent 30+ hours a week really trying to understand the material and working to do the assignments.  I went at it with the honest belief that if I just kept working at it, I would get it.  In addition to the office hours, I also tried to get tutoring from the school (was not available) and asked friends for help with concepts.  In exams, I scored in the 10-20% range too and it was incredibly frustrating.  I spent a lot of time trying to work things out and during that time, when I would get really upset, I would wonder, "If I'm spending this much time and energy on this and no matter what I'm doing I just can't seem to get it, maybe it isn't me."  Looking back, I think I can honestly say that the problem wasn't solely mine---while the professor had good teaching reviews, the number of people who dropped the class was around 50%.  Additionally, I'd like to think I'm not a complete idiot and I did finish both my graduate degrees at a top ten institution in my field.
The reason why I bring this up is because there were a few things you mentioned that bothered me a little.  For one, you said she sits in your office hours every week for three hours.  Maybe she is looking for excuses, but sitting in that many office hours is probably not her idea of a fun time.  You also said she does well in the homework sets.  I'm curious about how much time she's spending to do well in those.
You definitely have more perspective on the situation than I do and as I said, if she's got an entitlement issue and is making your life miserable, then something needs to change.  However, I just wanted to say as someone who was a struggling student in a class that had many of the same issues that you list, it was kind of painful to see you asking about what you can do to make her drop the class.  It was also painful to see the question about limiting the office hours you dedicate to her.  Yes, it shouldn't be a rehash of the lecture, I just wonder if the three hours of office hours are just the tip of the iceberg when it comes to the amount of time/effort she's putting in to really trying to get the concepts.
That being said, it might also be a complete mismatch of communication styles.  It may be possible to try having an in-class exercise asking students to explain the concept you just taught to the person next to them to see if she's more able to understand that way.

Wonderful comment and this is something I have experienced first hand. Not only did was I shoved into a graduate level course in a totally different department, but on top of that I was missing a few important prerequisite. I went to all office hours, all TA hours, I worked day and night on homeworks and scored poorly. I would like to blame it solely on myself but I just cannot, because I realized when the professor would start on a brand new topic that does not require those missing prerequisites, I would earn top mark in the following homework assignments. Close to 70% of the people dropped

And it was already considered an open ended course in the math department (I'm in a completely different department), the professor's style was to make it even more open ended by asking questions so hard the TA who nearly has a PhD in his field could not come up with answers. More frustratingly after I have done the course, I compared his grading style and assignments, tests with previous years and found that previous years was much more reasonable and the course was much less open ended. So now I am a much more sympathetic person to students who are not performing well.

Just a few added points.

Make sure that this student understands how the grade will be generated, and where she currently stands.  If this student isn't failing, you should put your grading scheme under a microscope and figure out why.
Make sure that this student has all the prerequisites for this honors class, and really belongs there.  If she doesn't, recommend that she withdraw, and then start up a policy of reviewing prereqs before letting students continue in the course.  If she does have the prereqs, take a good hard look at whether the other students have better backgrounds, and up your prereqs in future iterations to match what students need to excel in the course.

Enforcing prerequisites can be harmful to non-traditional students. I started my PhD in computer science in 2002 with my only formal CS education a master's degree completed in 1975. It would have taken years to do all the chains of prerequisites leading up to the graduate courses.

@PatriciaShanahan That's why they ALWAYS say "or instructor''s permission"

I am going to suggest a rather different approach. Others have mentioned that she should use a tutor, I am going to suggest that you should be the one to go see a tutor. Specifically I am going to suggest that you have a one-on-one with the head instructor of the tutoring staff to get advice on how to deal with this student.
I assume your institution has a certified tutor training program such as CRLA and whoever is in charge of that program is the one you want to talk to, their title may be something like Learning Specialist.
We all have our fields of expertise and sometimes it is easy to forget that tutors do too. The job of a tutor is not to help students do the homework, but rather to help the students to become better at being a good student. You would be wise to get advice from such an expert. Perhaps they might even be willing to observe a session with your student in your office, or you could prearrange to have the student meet you and the tutor lead at the tutoring center if you have one on campus.
Doing this may help the student to survive not only your course but many others as well.

For the benefit of those at U.S. universities who might think that such a program does not exist at their university, names such as "Learning Resources Center" are what to look for.

A little confrontational, but otherwise a good point.

@jpaugh This wasn't meant to be confrontational. When I suggested that the OP see a specialist it is because it is obvious the OP cares and is looking for good advice. We all have our opinions and classroom experience but a learning specialist is the best qualified to help the OP to help the student. These people focus on one thing, helping students, and they know more about pedagogy, androgogy, and the wide spectrum of problems and solutions in that field than any "mere" professor :) who must remain focused on other subject matters.

This sounds like she need a little more professional help than could be given by a professor. Universities sometimes haves offices where there is free counseling available for student-related issues, such as dealing with anxiety (which may be the reason of the discrepancy between her test and homework grades) and the difficulties in getting more self-dependent.
You may have to sweeten this suggestion to her by a (truthful as I suppose): Your homework grades are excellent, but you require a lot of help to do it; and in the tests you seem to have some problems; maybe you plainly are nervous or want to do everything perfect. Currently I don't see that you will pass the course, but make sure you don't fail again for the wrong reasons.

There are two ways how to deal with it.
If You want them to pass You can suggest them to look for tutoring. Suggest them to ask their schoolmates or older students for the tutoring. Your assistant or PhD. students are option too.
If You are neutral about them passing your course You can show them the tests results and suggest them leaving the course. If it is in the start of the semester, there sould not be many obstacles to switch courses without punishment.

I wasn't great student and I was lost in one part of Thermomechanic course. It consisted of one lecture and one practise per week. I asked the PhD student that conducted the practise lessons for tutorial. During this tutorial they taught me everything I needed to understand the troublesome part. They just found the way I was thinking and explained me the topic this way.
I also selected one lecture, optional in my studying plan, and after first lecture I talked to the professor. I told him I had been totally lost in the lecture and that I hadn't know it was final lecture for material science. He suggested me to ask for lecture withdrawal and selecting different lectures if needed.


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How honest should one be with their students when talking about the realities of academia?
January seems to be that time of the year that I ponder this question often. After the grades have come out, I always get a few students with very low grades (D or lower) coming to talk to me, and wondering whether this grade will have an impact on their future ambitions of going to a top graduate school and becoming a professor.
Another related event that happens in January is that it's the interview season for tenure-track positions in my field. The market is getting increasingly tougher, and plenty of very strong postdocs that I know don't have any interviews.
Watching these things always makes me wonder if it's a bad thing to be discouraging to the students.
Sure, technically it's true that even with a couple of failing grades, you can still gain entrance to a top graduate school, given glowing recommendation letters. And sure, I'm sure that some researchers weren't phenomenal in graduate school but they flourish in their later career.
However, in today's tough market I think that these things are becoming rarer, and in general, unless you stay on top of the track you don't really succeed in becoming a professor/researcher. So I often wonder if it's better to tell the students a bit more of the honest truth. That their ambitions are possible to realize, but that the competition is tougher than they could ever imagine. That they should definitely have a plan B in place. That even if they succeed in becoming a professor, their life might be very different from how they imagined it to be (struggling with a two-body problem, working for a university that you never dreamed of, and so on). I, for one, might have appreciated an honest answer, and maybe that would have changed my career trajectory. And I think enough people with doctorates work for jobs that they could have gotten before going to graduate schools.
So, as an educator, it is bad form to be discouraging. But I wonder if it's actually in their interests to really know what's in store. Do you do this? That is, do you tell your weak students with unrealistic goals that their goals are unachievable?

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While not an answer, I think this is very closely related to the "idea" that many in academia have, that "Your worthless if your not sporting a 4.0" I have seen may people fall into the trap of "You must do X at a university or you can't do Y." Enough so that brilliant inventors stop inventing because they can't get a perfect grade in the engineering class. So I think what ever you decide to do that you keep in mind the balance between telling the truth (important) and letting them explore their goals.

This is more important when talking about "teaching only" professions. So their goal is to be a professor, and it's true that they should be aware of the difficulties involved, but part of going to school that a lot of people foret is self discovery. And I see a lot of students stop that process because of 1 or 2 poor test scores.

One bit of bright news... if the goal is research or professorship, the amount of research done in the world is increasing, even if it's not as often within academia. Therefore, from a goals perspective, the difference between a top graduate school and a middling graduate school is shrinking for this purpose.

I am a student so this is coming from a student's perspective.
I think you should be honest with them regarding what going into graduate school entails. For my case, none of the professors I initially talked to encouraged me to go to graduate school, and for good reason. Each of them stated what the cons were to going to graduate school and how the academic job market is now much more competitive than when they were a fresh Ph.D. graduate. Was this discouraging at first? Yes. However, it also helped me in cognitively organizing why I wanted to go to graduate school and I was able to set a firm goal and stick with it.
Those professors often told me that the reason why they would encourage students to take a gap year and think long and hard about the next step is precisely because students often make the assumption that Ph.D. is the obvious 'next step' when it very well might be the exact opposite of what they wanted.
Coming from a student's perspective, I appreciate honesty from professors because it helps me to realize the reality of going into graduate school and potentially what kind of sacrifices I need to make (e.g. a good chunk of my youth, vacation time, etc.). However, I never found it appealing when a professor, who I aspire to be like, tells me that I'm not good enough to ever do what they do without additional feedback.
In my case, when I first approached those professors about my plans to go into Ph.D. to become a professor, my main advisor looked at my records and told me what to expect, where I currently stand, and what I can do to improve should I wish to stay on that path. I think the last bit is the most important piece of feedback I received. The first two parts can help me in deciding if this is the correct next step for me, but the third part is what will help me achieve that step despite the challenges.
So all in all, if I were one of your students, I would appreciate honest commentary on my goals as well as the additional feedback on what I can do to improve so that the choice of pursuing a Ph.D. is still ultimately my decision. This also helps students to develop and mature, I think.

Your first paragraph oddly reminded me of Starship Troopers, where the recruiter was a sergeant (one arm, no legs) whose job was discouraging recruits to become soldiers/citizens, but if they willingly joined and truly believed they could cut it, he'd be glad of having done his job properly, almost as if it was an accomplishment.

While I don't have a definite and complete answer, there are a few points I would like to make.

I consider part of my duty to help students get the right information to choose their path, and in France, it is actually listed in our official duties (this is relatively recent, was added less than a decade ago). So honesty is in order, really.
Before speaking the brutal truth, one should look into oneself for possible prejudice. One might overrate the importance of one's lecture because one feels it is the most important thing in the universe, while excellent grades in other lectures might make the student's record not so bad. One usually is influenced by common prejudice in our societies, and should be careful not to discourage female, working class or visible minorities students more strongly than wealthy white men. In some rare occasions, one might even find relevant to actually encourage some students who feel they can't do it while they are extremely bright and doing great (our students are usually subject to the same prejudice as we are, and it might be directed against themselves).
That said, given the toughness of the market to get a permanent position in academia, I now give the following advice to all but the very very brightest students, even if they do stand a chance: assume you enter graduate school, do a very good PhD, and then don't get a permanent position. Would you be ok with that, or would you regret spending these years in academia? If you can accept to move on after PhD, you may try it. If you'll regret it in case you don't land a job, I strongly advise you search another path. I give the same advice after PhD to young people deciding whether they should take post-docs (and how long they try): only take the post-doc you will be happy to have worked for itself.
It is also part of our responsibility, when we sit on admission committees or when asked to supervise a PhD (note that the system is different in France than in the US), to decline applications we do not believe firmly in. In France, there is relatively little recognition of the PhD in the private sector, and because of this I actually think we currently train too many PhDs.

+1 mainly for the excellent advice in the penultimate paragraph.

Um I don't agree with withholding the advice from the very very brightest students that you give to all the other students. They too should not be going into a PhD without really wanting to do the research mostly for its own sake, even if you're certain that they would 'do well'.

@user21820: there are some (rare) people to whom I could not give any other advice than "if you feel like entering PhD, just do it" given how obviously they should do great.

@BenoîtKloeckner: Well, as I said, it's okay if they want to do research, but it's usually not so okay if they think that PhD research is merely a stepping stone to something else. Rare or not, they deserve the same advice that you give others, since there is always a nonzero likelihood of not getting a permanent position after PhD.

I think you need to balance brutal honesty with treating students with respect.
If I find a student who will have a difficult time in getting into grad school, I would tell them that I believe they will have a challenging time finding a graduate school to accept them based on their current credentials. I would also attempt to explain what I think would make her case stronger, without being too harsh in describing the situation. You can also describe your own situation and the realities in the academic community as a whole in a similar manner.
Ultimately, it boils down more to how you say it, not so much what you say.

"I think you need to balance brutal honesty with treating students with respect." I agree, but I also think it is possible to convey one's full opinion of the chances of success in a respectful way.  It takes some forethought to choose one's words appropriately and some guts to say them, but it is certainly achievable with effort.

To echo @PeteL.Clark in different words... I don't think there is anything to balance; brutal honesty and respect are not opposites.

Did you mean to imply that brutal honesty and respect are antithetical to one another? Isn't being honest also being respectful? It might depend on what you mean by "brutal", but to me that part of it is more on a spectrum with "polite" than "respectful".

No.  You do not need to balance honesty with respect.  You need to balance honesty and your experience and knowledge with the fact that you can't predict the future accurately.  When I was teaching high school, I used to say things like "well, imagine an admissions officer. She has 40 applications to read in the next hour. After glancing at your transcript and SATs, is she going to read the rest of your application?  Maybe, but you can see you'll need a Plan B."

So, as an educator, it is bad form to be discouraging. But I wonder if it's actually in their interests to really know what's in store. Do you do this? That is, do you tell your weak students with unrealistic goals that their goals are unachievable?

As university teachers, we have the luxurity of working with adults who can and should make their own decisions. As such, it inherently feels "wrong" to me to present a student with a more rosy view of their situation to keep them on a track that you suspect they would have already left if they had all the information. It's also important to note that the short-term "encouragement" the student may currently feel if you tell her/him that (s)he may still get into MIT will turn into bitter disappointment and discouragement once (s)he finds out that (s)he won't get in. Even though as a teacher you should be encouraging, you don't necessarily need to choose a path that's maximally encouraging right now, without any considerations of future prospects.
Edit:
Regarding being honest with students about the realities of life in academia I would say you should definitely do that. However, don't just focus on the negatives - also explain why you like doing what you do, and why you think it is right for you. I think there is currently a trend in academia to be overly negative with everything, and I don't think that helps anybody. I feel the best course of action for me was always to be very honest with my students on what I am currently working on, applying for, and struggling with. That way my students get a good, direct view on what life looks like for a junior PI, and can themselves decide if that is something that they want for themselves or not.
So I think that ultimately, we do have an obligation to be honest to students about how we we see their prospects.
However, there are a few things to keep in mind:

You should not just tell students that their current dream is unachievable, but offer them useful alternatives to think about (and offer some good, plausible encouragement why the alternative is also very worthy). A student may not realistically go to MIT, but maybe a less excellent school is still within reach? A recent graduate may currently not be competitive for a tenure track position at an R1 university, but would they also be interested in a postdoc or a temporary staff scientist position? These things may seem obvious, but sometimes people get so hung up on their "dream" first choice priority that they forget that alternatives exist that maybe would not be too bad for the student.
You should always critically examine if you really have the student's best interest in mind, or your own. It's not important whether you consider teaching at a community college as a step down from graduating from your institution - if you can see the student be happy in a pure teaching position, you should encourage her/him to explore it even if it would not be the right choice for you. Similarly, you should not think about your tenure or promotion case while deciding about what to tell your student about his future prospects.
Choice of words as well as timing is of tremendous import if you want to be encouraging. Make very sure that the idea that you transport is that the student should be persuing the second-best option, and not that the student failed and should now be scrapping for some sort of semi-viably backup plan. Whatever choice a student makes, frame it positively and don't focus on what (s)he has not achieved to do. For instance, if a recent graduate will leave academia entirely for industry, your conversation should circle around what the student has learned and achieved, and not on that "he could have gotten this job also without a PhD". A good part of being encouraging is simply to be a "glass is half full" person.

Re: "You should always critically examine if you really have the student's best interest in mind, or your own": This makes it sound like professors have a conflict of interest (that they might benefit in some way from having the student receive this advice); but judging from the rest of that paragraph, I guess what you mean is something like "You should always make sure you're correctly putting yourself in the student's shoes -- every person is different, and the student's wants and needs may not be the same ones that you had at that age." Do I have that right?

One thing that is present in this that I always believe in: encourage students to think of the goal not the role.  Being in an R1 is a role. Being able to change the world in  way is a goal.  Money is seldom a real goal.  Having enough money to  is a goal.  I've interviewed a lot of people, and you would be shocked about how few can answer the question "If you could change one thing about the world with you work, what would it be?"  So busy trying to get there, not sure what to do if/when they make it.

Totally. We're talking about years of someone's life, here.
By way of elaboration, let me tell you a story. I applied to postgraduate research positions at three universities after I graduated. One was rejected. One, the one I wanted most because it aligned most closely with my research interests, I didn't hear back from. The one I wanted least I was invited for interview and got the position.
Several days after accepting I got a letter from the institution I most wanted to attend offering me a place. The professor apologized for the late reply and offered some excuse (I forget what). But he said he knew one of my tutors and was happy to accept me on the basis of personal recommendation. Because I already had a position and it was literally days before the start of a new academic year, I felt honour bound to stay with what I had already accepted.
During my first few months of postgraduate life I discovered that my tutor was seen as a liability by the rest of the faculty. He was ill-tempered, physically threatened his students and had almost zero concept of how modern research was conducted. He spent most of his supervision time with me telling me off about minor errors of grammar and spelling in my lab reports. When challenged or asked to provide scientific guidance he would grow increasingly angry to the verge of violence.
I asked various people who'd been part of my interview process why they hadn't warned me when they had ample opportunity. They all apologised and said they were all too scared of the professor in question to risk telling me the truth.
I flunked the course after two and half years of misery with almost no material for a dissertation.
Truth be told, I might have flunked any PhD course. It may be I just wasn't dedicated enough or good enough at benchwork (my theory was fine!) to make the grade. But I wasn't given the opportunity to find out whether that was true because I was lied to about the quality of course I'd enlisted on. Those lies cost me two and half years of depression and the chance of a doctorate.
Even if I'd had a better tutor, they could have done me a huge favour by warning me upfront that academic life is hard. That it requires lots of out of hours work and enormous dedication. That it requires skills you've likely only touched on briefly as an undergraduate and that it's totally different to undergraduate life. That even an good to average student might not cut it in this world. That postgraduate life is structured and administered differently from undergraduate and that there are far more personality clashes and far fewer quality checks.
I might not have listened had a nicer tutor told me these things. But at least I would then be certain I only had myself to blame. Failing to be upfront about the academic life and, specifically about the academic life at your institution can cost people years of their lives. You own to them to be totally, unequivocally, honest.

One thing I don't see mentioned yet is that you might be able to provide some hard data for the student. Statements like "academia is very competitive" are vague, because people have different ideas of what competition means.
Suppose a student really wants to be a research professor at a top-10 or top-20 school. In their past they have been in the top 10% of their class in high school and the top 10% of their class in college. They think they should be able to pull off being a top 10% grad student as well. This person has been very competitive and academically successful their entire life, so telling them that the job market is competitive doesn't really convey useful information. They might not realize that the position they're shooting for will have them competing against the top 1% in the nation (and consequently, they might not realize that academic scores don't mean a lot up at that high level).
In my field, Computer Science, we have a publication called the Taulbee Survey which gives very good data on all of this. According to them, the field produced 1,780 PhD's last year. Of those, just 140 got a tenure-track position in a PhD granting department in North America. That's enough of a baseline to say that only 10% of PhDs end up as research professors in my field. If a student wants to be a really high-power research faculty then they're going to be competing for something like 20 possible slots in the US.

And I should point out that CS education is booming right now. I would definitely inform your students fully if they want to go into one of those fields with just a handful of openings every year.

If you sum up all academic employment, you get 473 (or so), which is about 1/3 of people for who they have data (1406).

The 10% figure is a low-end estimate. If not all the 1780 new PhDs sought tenure track prof positions, the percentage is higher.

I'm also a student.
Be very honest. You have information that they don't have that will be useful for them. I would rather tell them what you know and have them make the decision for themselves, rather than let them keep doing what they're doing and seeing them fail.
A quick personal story: I was an undergraduate student about two years ago, getting top awards in the math department that I graduated from, was well-respected in the department, and told my advisor at the time that I was thinking of pursuing graduate school full-time after a gap year of working in actuarial science, to eventually become a professor of statistics.
My advisor was involved in the hiring committee at the time for the math department, and she had a lot to say about this. To paraphrase what she said to me: the market is extremely competitive (and I understood what this meant, considering that I previously was considering to be a musician full-time), she was disturbed at how professors seem to always suggest that their top students go to graduate school, and she strongly suggested that I spend some time exploring my options in the job market.
This was the best advice that I'd ever had in my four years as an undergraduate. I did what she said, and yes, I did have my struggles. But I am very happy with my life only two years later, including the fact that I'm working full-time in a job I love while working on a graduate degree part-time. I hear from my peers who are in Ph.D. programs full-time who are considering dropping out, most of whom I consider to be very intelligent people. I hear about the things that they are doing in their programs and their morale, and I prefer my life of being able to pay for my graduate-school expenses and developing professional experience over the life of being a Ph.D. while TAing courses of students that do not care for the material.
Dreams need to balanced with reality. In my college career, I wanted to be a full-time musician and a full-time professor. It is easy to be lost in what you're doing if that's all you're doing for 40+ hours a week, especially if you're a student. Students who are spending many hours on their studies and other obligations often don't have time to be doing their own research on the job market itself. In most cases (judging by my peers' experiences), they often don't know where to start.

I would be honest that low grades will make it difficult (though not necessarily impossible) to get into their dream grad school. And in fact getting into such a school and not being up to the challenge wouldn't be good for the student.
If the student insists on wanting to go straight into a doctoral program, I would suggest that they apply to some 'back ups' as well as their top choices.
But I would also advise them to consider doing a master's degree program that is rigorous enough to prepare them for a Ph.D.
program later.

Having thought about this issue greatly for both mentoring and as a research question, I believe the solution is: quantify it.  Answer in terms of time, effort, and personal sacrifices.  Those are really the trade-offs.
It's relatively straightforward to have a student review some CV's of recently accepted grad students, or whatever other goal position that a student has. It can then be quantified a bit: how many publications are they behind?  How many awards? How many skills/fundamental courses are they behind their competition? How much studying would it take to get their standardized test scores at or above average? How good are their writing skills compared to the average?  Unless a student can demonstrate themselves to be certifiably brilliant even in brief conversations or through some singular achievement, they're going to need a good body of work. This is not a great measure for excellence or slam dunks, but it is definitely a good measure of being able to have a fighting chance.
At that point, it is possible to estimate the amount of time that would be needed to reach the average level of the competition who is being accepted.  Are they already around the average? Three years behind the average?  At the point where they are behind the curve, it is reasonable to ask if they are willing and able to do a regular pace of 80h+ weeks over some period to reach an average benchmark (if sufficient time even still exists for their CV to make weight even with unwavering effort). If they have already been pulling 85h weeks and they're still behind, I would say to go for it (they're obviously determined to) but to have a good fallback plan.  If they're not yet maxed out, it's a good time to have them re-evaluate their effort and grit.
The kind of effort needed to play catch-up in academia is fairly brutal, which should also be communicated in terms of personal sacrifice. Social outings can be scarce, family commitments limited, relationships stunted or sacrificed. Work-life balance will become a Globetrotters-Generals match. Perhaps worse, if they do catch up and can keep pace (but at that effort level) they will then be hired on the expectation of results and corresponding effort level. Or, put simply, they could be working 80h+ weeks for the rest of their life. I have to imagine that divorce rates go up at least 1% for each hour/week over 80h, as well. At any rate, if they're going to reach, they're going to need to love it- because there may be little time for other things that they love.
So I think there is no need to discourage them. Be honest. Ask them to be honest with themselves. Let them set their goals and track their goals. When confronted with a barrier, some may strive to overcome it and others may change course. Both options are acceptable: that is their choice. But they should be shown the hurdle before they crash into it. A few of them might surprise you (a Rudy). Most won't surprise you, except perhaps in how quickly they lose pace on their new regime of extra effort.
Note: One exception to this rule is if your field is so overfilled that PhD students do not receive fellowships and go into debt to complete them. In that case, tell them to run (don't walk) away from graduate work in your field. If there's one great rule, it's that if people can't afford to pay a grad student a modest stipend to do research, there's almost certainly not any decent jobs on the other side either.  The one exception to this one exception rule is if they're independently filthy rich.

Also, requisite watching for students trying to really reach might also be the film Gattaca.  At a certain point, you're talking about using everything you have to get somewhere.

"Or, put simply, they could be working 80h+ weeks for the rest of their life." No, only till tenure. Assuming they can get tenure, of course. And another, not uncommon thing that helps people achieve success in academia, is favoritism/nepotism, and preferential treatment. But those are relatively fringe phenomena.

"At any If they're going to reach, they're going to need to love it". This sentence probably has a missing word. And that "If" shouldn't be capitalized.

Fixed that.  Was a shift from a straight "If" to an uncompleted "At any rate, if..."

Also, from what I have seen, at the point where you have worked 80h/week for over 10 years (i.e., graduate school through tenure), it's a lifestyle. Some might ease off, but most of the ones that I know just as many who keep their foot on the gas. On the other hand, one wouldn't run into the others as much anyway, I suppose...

This is a great answer - wish I could upvote more than once.  The lifestyle and sacrifices mirror what I have read and seen from others including close friends of the family.  One of them was salutatorian from Maryland in Math, got their PhD but never a tenured position due to a nearly complete lack of positions available.  And the hours invested and life experiences missed..

One thing I didn't mention here is the difference between doing a PhD for the role (e.g., professorship) rather than for the skills (e.g., apprenticeship). If you want the skills and need the skills for some goal that you can control, like taking a whack at world hunger, you've got a lot of latitude to turn even lemons into lemonade. For example, with a math PhD, you could technically spend time solving equations in your free time, like https://en.wikipedia.org/wiki/Christopher_Langan who was a bouncer. If you want the role, that has a lot of risk due to limited slots.

Or, TL;DR version: roles are socially-assigned and there is limited control over them. But a resourceful person can make progress on the same goal through a variety of roles.  And if all someone wants is the role with no ultimate goal, that really requires re-examining one's motives for completing a PhD.

While being (reasonably) honest is certainly important (offering a hopelessly pessimistic view may be honest, but not reasonably so), I would like to emphasize an aspect that does not seem to be covered in the answers. The OP is an advisor to students, so, obviously, one of his/her main functions is to advise them:-) An advisor does not have to give a yes or no answer even if the question seems to require that. There seems to be a problem with students that they often do not know much about careers outside of academia, and this is often a (bad) reason to have an ambition to become a professor. What is worse, their academic advisors often don't know much about non-academic careers either, as they often spend their entire life in academia. It so happened that I did not find a position in academia after my post-graduate studies and started to work in industry, and it was quite interesting, view-broadening, and requiring much of the skills I acquired gaining my PhD. After I worked there for 6 years, I was invited to work at my alma mater (based in part on what I was able to do in the industry) and worked there for another 6 years. Some people here complain that they lost years working for their PhD, and I am not sure that they just wasted their time. PhD is a step in our studies, and the skills you acquire while studying are typically applicable in many areas. So maybe advisors should also tell their students that the latter study to prepare for life, not just for academia, and they should try to gain a broader view of their future careers. People mentioned Ramanujan here. I would like to mention Tsiolkovsky (https://en.wikipedia.org/wiki/Konstantin_Tsiolkovsky) and Grassmann (https://en.wikipedia.org/wiki/Hermann_Grassmann), who worked as school teachers.

Very simply, tell the students the truth.
When people enquire with me about doing PhD, I always tell them that it is "blood, sweat and tears", using exactly these words. This is not entirely true for the PhD in general, but it is true if one wants to excel in its field.
Excelling during PhD is necessary condition for them to be even marginally employable in academia (as you are noticing, the academic market is increasingly tightening), so this may be relevant to your question.
I also sometimes tell people that the PhD is like they have been cruising casually until now, but with PhD they will ride a rocket into an asteroid belt, with me helping to avoid the asteroids, but with them actually steering the rocket.
You may find those two analogies helpful :) It may weed out the people whos PhD would be a waste of time.

Yeah Its Coming from a student's perspective, I never found it appealing when a professor, who I aspire to be like, tells me that I'm not good enough to ever do what they do.

Trust me, professors don't find it appealing to say discouraging things to students, either. However, academia really is very competitive, and if there is a reasonable chance (say 99.9%) that a student will fail, despite the non-appeal, perhaps there is some value in discouraging (but not forbidding!) the student anyway.

It's not about you. Today in most fields, if you're an excellent postdoctoral you still have 1% chance or so of getting a professor job. You may resent it, but the only brutal part of the story is that several current professors would not stand a chance to become professors under nowadays competition.

Is that what they say? Or is that what you hear? Personally, I would find a remark such as, "You're not good enough to ever do what I'm doing" to be in poor taste. However, something like, "You're going to have to develop better work habits to make it in this career field" might be a fair warning to a student who consistently turns in shoddy work.

This is a great way to send an ultimately unsuccessful prospective medical student on a two-year scenic route. What I needed back then was the truth, not sweet talk. Things are great now, but I shudder to think about what my life would be like had I actually made it in only to flunk out later.

"Not good enough" isn't the problem.  The problem is that there are not enough academic positions for everyone that is "good enough", or even for everyone who is absolutely brilliant.

@famargar: "Today in most fields, if you're an excellent postdoctoral you still have 1% chance or so of getting a professor job."  Come now.  The job market is very tight, but let's not make up statistics.  I'll show you statistics for my field and country; see here: http://www.ams.org/profession/data/annual-survey/2014Survey-NewDoctorates-Report.pdf.  We graduate (numbers are rounded a bit) 2,000 PhDs per year, and about 650 of these do postdocs.  We fill 750 tenure-track positions per year: http://www.ams.org/journals/notices/201609/rnoti-p1057.pdf.  One in 100 is clearly wrong.

"However, I never found it appealing when a professor, who I aspire to be like, tells me that I'm not good enough to ever do what they do without additional feedback."  This is an exact quote from the accepted answer.  Which came first?

I suspect that this question hasn't been asked by a professor in social science. I say this because I think the OP has little regard for the various factors and complexity which may influence a student's track record.
A student may not perform as well currently, in a certain university and country, doing a certain study, in his current socio-economic and medical situation; but may perform staggeringly differently in another time and place.
Further more, what "top graduate schools" are, is a matter of debate and continuously changing.
Therefore, if you want real intellectual honesty, one should admit to your badly-performing student that you can not predict his future academic track with exact certitude, simply because there are so many chances. Keeping this in mind, there are more diplomatic and intelligent ways to encourage learning in your student's mindset, e.g. telling him that the better he wants to be, the better he will become. His studious attitude may even be rewarded 20 years later when he starts to study Y instead of X. The student might even re-start the same study X 30 years later: you simply cannot know.
Treating your students as if you are 99.9% sure of what they will become seems like treating your students as if they were inanimate objects, of which you can calculate the trajectory using equations from an exact science. I think this way of thinking about students show that professors who teach, should actually get to know something about pedagogy. Because one can know an awful lot and be an expert in his/her subject, but have no clue about good ways of teaching (on a pedagogical and motivational level).

"the better he wants to be, the better he will become" oh $deity, how I am happy that my mentors and advisors usually gave me something a little bit more tangible than that.

the better he wants to be, the better he will become — This is absolutely horrible advice.  The better he works at becoming, the better he will become.

This kind of answer reminds me of physicians who are unwilling to speak in probabilistic terms under any circumstances.  It is really unhelpful, and the necessity of it is clearly falsified by other (better?) physicians who will happily speak in those terms.  Do you really feel that there are no circumstances in which your own past experience gives you more insight into the student's likelihood of attaining a future academic career than the student presently has?  I find that rather frightening.

@ JeffE I agree, but you will agree that no one works without the will to work. So, we agree that our advice is not horrible. @Pete L. Clark Medicine isn't quite the same as social pedagogy, but I appreciate your analogy. My point is that students, as well as academia, are inherently social phenomena. There must certainly be geniuses who are e.g. inhibited by some social factors at some point in their lives, but perform brilliantly once these factors have been eliminated at another point in time and place. Nevertheless, I can agree with predicting in some mentally retarded cases of course.

@Vincent: I'm afraid I don't really understand your comment, and not because I have any trouble with academia as a sociological construct.  (As a practicing academic, I am very well aware of this.)  "There must certainly be geniuses who are e.g. inhibited by some social factors at some point in their lives, but perform brilliantly once these factors have been eliminated at another point in time and place."

Of course there are, but so what?  Are you saying that any student you interact with may be a "genius" -- or, more soberly put, someone whose potential is dramatically hindered by contingent, temporary factors -- so that they could do much better in the future than they have in your observed interactions with them?  Of course that's true, but that's why we're talking about probability (one is virtually always only talking about probability when giving advice or making any decision whatsoever).  I somehow doubt that you ignore probability in your own life, so I don't get your position.

Sure @PeteL.Clark one who is an expert in the probabilistic theories of academic success might, to some degree and with some error, be able to predict chances in some general populations and so on ... This may be based on population studies. I think we both agree on that. High income of parents might be such an influencing factor? Perhaps. But I feel like my answer is a necessary antidote to the OP. See e.g. what he/she says below: a reasonable chance (say 99.9%) that a student will fail. I am saddened by this claim of 99.9% predictability on an individual.

I wonder if the OP also has a crystal ball to predict reasonably accurate (say 99.9%) which students will "succeed"; which is of course also impossible.

@VincentVerheyen I think you're misinterpreting some logical quantifiers in my statements. I can't predict success for all students, but there exist some students for whom I can predict with 99.9% certainty their failures (lazy, lots of F's, no initiative, etc.)

@Sana Amazing that you think that people might not change in their laziness, or in their F's, or in their initiative? This look at students is exactly what I am opposing in my answer. You have to be aware of the fact that what you say actually can change a student's initiative and laziness: again, students are not static entities.

@Vincent: 99.9% is indeed a high chance, higher than I would report for most things.  I will be honest and say that I have never had a student approach me about an academic career for whom I thought they had only a .1% chance of academic success.  But I don't see this justifying assertions like "Therefore, it is always imperative to explain to your student that you can not predict his future chances, simply because there are so many chances."  If I may: I suggest that the antidote of exaggerated claims is not exaggerated claims in the other direction, but rather accurate claims.

@PeteL.Clark Thanks, I have added some nuance to the formulation.

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