I was once told that half of Nobel laureates were the students of other Nobel laureates. ... Even after discounting for cherry-picking of students and political pull, this suggests to me that you can learn things by apprenticeship - close supervision, free-form discussion, ongoing error correction over a long period of time - that no Nobel laureate has yet succeeding in putting into any of their many books.

What is it that the students of Nobel laureates learn, but can't put into words?

You can't put mentornship in a book. When I face a problem that may or may not have a solution I find it useful to convince myself that there is a solution, and that I only need to find a path to it. Once I eliminate the doubt or fear that I might be wasting time I'm able to concentrate on the problem at hand. If you define "success" as a problem that may or may not have a solution (ie. you may or may not be able to achieve it) then studying under a super-star may give you a psychological edge over others in the same field. It's a form of tacit permission by which you subconsciously feel entitled to success and may be more likely to take gainful risks or less likely to simply give up.