Change- and unchange-of-state 了: During which time in a person's life is it correct for them to say 我三岁了 (I am 3 years old)? 我三岁了
  I [am] three years of age


This breaks apart as follows:


我 (wǒ) = me
三 (sān) = three
岁 = years of age
了: I understand 了 here is the change-of-state 了: prior to turning 3, I was not 3, so that state has changed.


I have two points of confusion regarding when it's correct to say 我三岁了:



Two years old: 我三岁了 is incorrect because... (why?).  It seems correct to say e.g. 明年我三岁了.  It seems theoretically possible that 我三岁了 could refer to a future change of state (although surely none would use it because it's misleading).
Three years old: 我三岁了 is correct: I turned 3 (change of state).  [more about this in point 2.]
Four years old: 我三岁了 is grammatically incorrect: my best guess as to why is because while I turned 3 (change of state), I subsequently also turned 4 (unchange of state). With this reasoning, 我去年三岁了 is self-contradictory (maybe?).  And Baidu indicates it's correct to say


  我去年三岁
  Last year, I was three years of age


without the change-of-state 了.  

When I'm three years of age, for that full year, I'm not sure if it's okay say 我三岁了.  For example, I wonder if it's okay to say


  我三岁了，明天我四岁了.
  I [am] three years of age, tomorrow I [become] four years of age


The change of state happened a long time ago, so I'm unsure if it still counts as a change of state (but maybe a change of state has no expiry).


Question: During which time in a person's life is it correct for them to say 我三岁了 (I am 3 years old)?
Correct for "change".


Usually say 我三岁了 on the birthday of three. That's the day of change.
When four years old, remove 了 as past not change.
On the birthday of four, we usually say 我三岁，明天我四岁了, as there is no change for the first part.
Any other cases for "change" during the year of 3? Yes. When someone asks the little cute toddler that "r u 2 years old", the cute will reply nooooo, 我三岁了 proudly, to indicate the "change".