the best mathematicians I work around tend to have gone to ordinary good universities and distinguished themselves first. then they were invited to the top universities.
the other direction, getting into top universities first, is a big advantage, but i would say it confers competence rather than greatness. I.e. almost everyone that trains at a top place is competent afterwards, but not necessarily outstanding.
the advantage it carries is exposure to outstanding people, and that gives one a goal to shoot for. also one hears top quality instruction, but it still remains to do something with it.
if you want a story of ordinary success rather than greatness, i am myself an average garden variety mathematician. i went to a top college, wasted much of my time, and left with little to show for it except the experience of having "been with saints", so to speak. I bounced down the professional ladder of mathematics until halted by landing near the bottom (a temporary job at a state college).
Then I became happily married, a father, and highly motivated. I attended whatever grad school offered the best support for my family and worked my *** off for 10-15 years. After 9-10 years I was invited back to the same top college I had bounced out of.
Interestingly, a world famous professor at this school once peered curiously at me and asked where I had come from? I said I got my PhD at [state university], but he persisted, " no where did you go to college?"
When I named his own famous university, he said "oh , you are a *...* man!" so he at least chose to believe my original exposure to his school caused all my subsequent success. I found this simultaneously flattering (that he thought I had accomplished enough to pique his curiosity) and insulting (that he gave me none of the credit). To what, I thought, did he attribute my years of struggle and difficulty?
This is of course not a tale of greatness or even excellence, but a tale of success wrung from failure. Things started getting good for me personally when i started working hard, wherever I found myself.
It does seem true however that really talented people who are lucky enough to find themselves also at top schools, do well. but recall einstein produced his Earth'shattering work while laboring in a patent office.
I have nothing to say about such remarkable people, but ordinary persons like myself are advised simply to do the best they can wherever they are, and perhaps aspire to more.
