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)?
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".