Can nerve damage repair itself?

No: If the nerve is simply bruised, then the nerve damage may improve with time alone. However, nerve lacerations or direct injuries often require surgical exploration and direct repair in a timely fashion. For chronic injuries, nerve grafts may be necessary.

Sometimes: A broad question, because there are many ways a nerve can be damaged: compression (carpal tunnel), crush, cut, etc. A "bruised" nerve can heal itself; it grows back at about 1 inch/month. If the nerve is divided, it may occasionally heal, but more often needs surgical repair--and this doesn't always result in return of function.