I think the truly great ones, across all sports and eras, are able to shrug off their defeats - or screw-ups - quickly and totally and move on to bigger and better things. I am not knocking those who are competitively destroyed by one devastating loss or performance. That is after all why the great champions - Nicklaus, Brady, Federer, et al - are so rare.
I read an interview with Arnold Palmer. He won only one US Open - no shame there, one US Open would be a career for many golfers - but was so close so many other times and never closed it out. He said it was so hard to maintain that indescribable edge necessary for world-class high-level competition, and you never knew how or when you could lose it forever.
Phil Mickelson and Louie Oosthuizen are experts :)
All of the truly greats say their losses and learning to overcome adversity was a big part of making them attain membership among the group of greatest.