A hat? Not likely. One of our Scouts made one out of tin foil this weekend so it's at the front of my brain. I keep a hat close at hand, if not on my head. How about fire alarm + blazing sun + ill co-worker? Leaky fire repression system after the nuke alarm goes off? Belief that the bosses are using mind-control devices?

That's one reason I carry a Buff instead of a regular bandanna. I have used it several times, but always as a clothing supplement - do-rag, watch cap, neck gaiter, balaclava, etc. I find the Buff is much easier to use in this capacity than a bandanna.
Here is a demo of the Buff for those unfamiliar:
http://www.youtube.com/watch?v=iQAKQ0y87d0