Nietzsche himself delivered a very dramatic demonstration that it's not true. What didn't kill him drove him completely insane and left him a helpless invalid for the last 11 years of his rather short life. He wrote many profound things, but "Was mich nicht umbringt, macht mich staerker" ("Whatever doesn't kill me makes me stronger") was not one of them. It's the last of 8 aphorisms on page 11 of the insel taschenbuch edition of Goetzen-Daemmerung, and it's the weakest, least authentic thing on the page. And to make the demonstration of its untruthfulness even more dramatic, and the whole case even more ironic, Nietzsche wrote that just weeks or months before his final, permanent, irreversible breakdown.
Nietzsche was in very deep denial about himself and the state of his health. It would have been much more accurate and honest if, instead of "Whatever doesn't kill me makes me stronger," he had written something like, "I am in very delicate health, and I must be very careful about what I eat and drink, where I go and what I do. And it would be a great stroke of good fortune for me if I were to meet a physician who is a genius -- soon!"
"Was mich nicht umbringt, macht mich staerker" ("Whatever doesn't kill me makes me stronger"), although completely false, is snappy. And so it has been one of the handful of sentences which have made him known to millions of people who have not read his books.
Nietzsche's breakdown occurred very suddenly, early in January, 1889. Still more irony occurs to me: In the last year before the breakdown, Nietzsche wrote and published a great deal, several books, full of swaggering lies about how robustly healthy he now was. He admits that he had had periods of poor health before -- but now, he insists, he is much more than merely okay. The irony is: what if he wouldn't have broken down if hadn't been working so hard on all of those brilliant books (they all contained much brilliance, besides the false swagger and denial about his poor health)?
What if "Was mich nicht umbringt, macht mich staerker" ("Whatever doesn't kill me makes me stronger") was actually the very last thing Nietzsche wrote in any of his published works, and was actually the last straw of effort which, although not killing him, utterly snapped all of his remaining strength?
Please, everybody, take good care of yourselves.