JFK made a deep remark when he said these words. I understand these words in two different lights:

First Interpretation:
"...........success that there were many people trying to claim their credit in it? On the other hand, when things go wrong, nobody usually wants to be associated with it."

I agree with this as it is the obvious and the direct meaning.

Second Interpretation:
I find these words to also imply that in a lot of cases success, comes from a joint and a concentrated effort. Thus a single event of success can be sired by many working together. There are also times when failure is at hand because nobody actually tried hard enough.

This second meaning is a bit of a stretch but nevertheless I find that it can be derived from the words of wisdom above.