i have a personal, not necessarily scientific opinion on this. i think that certain minds are so above normal life that they are able to creatively push the bounds of our knowledge. now, since life is a flawed beast of chaos and disorder, these minds that have high creative but yet logical horsepower are not able to deal with the shortcomings of humanity. they could also fall into this trap of "everyone says i am wrong, but they were wrong before when i was right, so it's the case again". i think it's these facts coupled with the fact that depression can be described as people who understand the world as it truly is but can't handle it or ignore it like everyone else.

have you ever dealt with someone so illogical it just drives you mad? now think if everyone you met felt like that.

i actually don't know much about godel, but his work on logic, computation, and general relativity seems to be of the flavor "the world may not be as you think it is". that type of work could take its toll, and something did.

A cursory search yields that actually mental health is correlated with academic achievement (and thus mental health problems must be anticorrelated):

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4443903/

Of course, this is "academic achievement" as in getting a 4.0 at Stanford, not creating a new field of mathematics...

I imagine your question would be hard to investigate as many of these high achieving mathematicians are rare outliers and would defy statistics on population averages...

Given the way mental disorders are defined, that's kind of unsurprising; adverse impacts to work, school, etc are required to diagnose many disorders.

Well, of course, Cantor, Gödel, Nash, Grothendieck, these are not everyday people.

What makes you think Grothendieck had mental health issues?

Among other things, in Lettre de la Bonne Nouvelle he described his encounters with a deity.