Is it possible for you to bruise a nerve?

My hand fell asleep: Yes and no. A bruise of the skin often is discolored, due to some changes in blood products. A nerve would not have this kind of reaction. A nerve can be damaged by trauma, and even mild damage can cause symptoms of numbness and tingling. Compression of an arm or leg that causes a feeling of "falling asleep" is due to nerve compression of this kind.