My sister had twin boys about a year ago. One of the boys had a nerve injury that keeps him from using his right arm and hand fully. I noticed I can lift his hand over his head but he can't do it himself. Why not?

You are noticing the difference between passive motion (you lift his arm) and active motion (he lifts his arm). There are usually two main reasons for a problem like this. First the nerve damage may have left the muscles of the arm weak.

Second muscles that don't move through the full range of motion can get tight or contracted. With limited use of the shoulder, the soft tissues around the shoulder can get contracted. The contracture can put pressure on the nerve in the area causing this weakness.

You've made an important observation. If you haven't mentioned it to your sister, do so now. Encourage her to have the child evaluated or re-evaluated if he was seen some time ago.