After recently posting that I’m using WPSuperCache, I had to turn it off. The .htaccess file manager wiped out the line that is needed for WordPress to do automatic updates on my hosting provider, and somehow managed to get itself in a state where the admin pages themselves were being cached. Nearly nothing worked, until I finally got the sense to backup the .htaccess files, disable WPSuperCache, and see what had been broken. The following line was needed to make the updates work again:

AddType x-mapp-php5 .php

Hopefully next time it breaks, I’ll think to look here.