Depends on how tight it is. I’ve hung upside down with a ball cap on and had it stay but I like my hats snug.
Washoe:
One time my father and I were hiking in the High Sierras, and we met a hiker at the top of a pass who was hiking in the opposite direction and stopped for a few minutes to BS. Somehow the conversation turned to electrical engineering, and when the hiker and my father realized that they both had backgrounds in E.E., the hiker offered that he had graduated from MIT. The hiker then suddenly realized that this could be construed as being boastful, and somewhat embarrassingly asked my father where he had received his degree, to which my father replied “Caltech.” The hiker paused for an uncomfortable few seconds, and then said in a hesitantly deferential tone “uh, well—that’s a good school too.”
I heard about two guys in the Boston area; one a Harvard grad and the other an MIT grad. The MIT guy would occasionally console his friend by telling him “Harvard is, arguably, the second finest university in the whole of Cambridge.”