OK, so I created a simple test file in Visio. No big deal. I'm done with it so I go off to close it, and get...
First, note the action: "Close Window." Next note the helpful suggestion of "closing and reopening the file."
OK, I can't close the file. So next up we have restarting Visio. Just hit the X to close the window, and...
Grr...
Turns out some macro in the background was taking forever to finish. Once I let it complete, I was able to close the file. So it's not Microsoft's fault, and I know whose fault it is (because they're also responsible for this and this), but I'll keep it to myself for now...