In order to do well in nursing school, you have to be willing to do a LOT of reading and a LOT of writing! You have to really be willing to put a lot of time towards school. You have to have good time management skills. All things that have already been said.
My belief however, is that you DON'T have to be of above average intelligence, by any means, to be a nurse, or be a GOOD nurse. The most important things are, again, to have good time management skills; but also it really helps to be someone who has good intuition and common sense. It helps to be someone who immediately can say, in a situation, "Oh well, this is what needs to be done" or "This is what I need to do" and then be able to DO it without too much hesitation.
My downfall as a nurse has always been that I THINK way too much and worry too much that in a situation, I might not know what to do or might not be doing the right thing. Not that some critical thinking isn't important in every situation that you encounter as a nurse, don't get me wrong!!! But I've seen nurses that seem to be able to think rather quickly and "on their toes", and be able to respond quickly and confidently in emergent situations, and that has always been a problem for me. It's the reason why I had to choose a slower paced type of nursing, with fewer emergent situations.