We will judge angels? But we are lower than the angels Know ye not that we shall judge angels? how much more things that pertain to this life?  1 Cor 6:3 (KJV)


This verse says that we will judge angels, but this verse 


  6 But one in a certain place testified, saying, What is man, that thou art mindful of him? or the son of man, that thou visitest him? 
  7 Thou madest him a little lower than the angels; thou crownedst him with glory and honour, and didst set him over the works of thy hands: 
  Heb 2:6-7 (KJV)


says that man is lower than the angels. Will we as humans, who are lower than the angels, in fact judge angels, who are higher than us? The citizen doesn't judge the judge; does the human judge the angel? And if so, when?
The simplest way to reconcile these passages is to dispute the KJV's translation of βραχύ as "a little." This is, in fact, what most other translations have done, including those that have no interest in internal harmonization. The NET renders Heb 2:7 as follows:


  Hebrews 2:6–8a (NET)  
  
  6 Instead someone testified somewhere: 
  
  "What is man that you think of him or the son of man that you care for him?
  
  7 You made him lower than the angels for a little while.   
  
  You crowned him with glory and honor.
  
  8a You put all things under his control,"


The NRSV, ESV, NASB, and CEB all translate this verse similarly, as "for a little while" or "for a short time"—a meaning well within the word's semantic range. The NIV is the only major modern translation that interprets βραχύ as a hierarchical rather than a temporal statement.

Here's the full entry from BDAG:


  βραχύς, εῖα, ύ (Pind., Hdt. +) prim. ‘short’.
  
  ①pert. to having little length, short, of space: βραχύ (so
  Thu. 1, 63, 2; 2 Km 16:1) διαστήσαντες_a little farther on_Ac
  27:28.
  
  ②pert. to being brief in duration, brief, short, of time: β.
  (τι) for a short time (Ael. Aristid. 13 p. 276 D.) Ac 5:34; Hb
  2:7 (quotes Ps 8:6, which refers to rank; in Is 57:17 β. τι denotes
  time), 9;**μετὰ β. a little laterLk 22:58.**
  
  ③pert. to being low in quantity, little, small (1 Km 14:29,
  43; Jos., Bell. 1, 597, Ant. 9, 48 ἔλαιον βραχύ): _a small amount_β.
  τι_a little_J 6:7 (cp. Thu. 2, 99, 5). διὰ βραχέων_in a few words,
  briefly_Hb 13:22 (besides the exx. in FBleek ad loc., s. also
  Just., A I, 8, 3; Tat. 41, 3; Ocellus Luc. 35; Ptolem., Apotel. 1, 1,
  3; Lucian, Toxaris 56; Ps.-Lucian, Charid. 22; Ael. Aristid. 13 p. 183
  D.; Achilles Tat. 7, 9, 3; PStras 41, 8 διὰ βραχέων σε διδάξω; EpArist
  128; Jos., Bell. 4, 338). LTrudinger, JTS 23, ’72, 128–30.—1 Pt
  5:12 P72.—B. 883. DELG. M-M.


As you can see, the KJV's translation uses the third definition, but even then it's a bit of a stretch. Based on the information here, βραχύς seems to be more a quantitative than a qualitative term. "A little lower than the angels" implies that we are created with a substance that is intrinsically lower in rank or quality than angels. I don't think the text suggests anything like that.