I ran through the usual troubleshooting procedures - disabled plugins, tried a different theme, etc. - to no avail.
I also tried inserting define('CONCATENATE_SCRIPTS', false ); in config.php but that didn't help either.
This is happening on multiple sites and each is on a different web host, so it isn't a server config issue.
I can add media to the gallery, but when I hit "insert" nothing happens. I have to copy the file's URL from the gallery and wrap it in img tags myself.
This has only been happening since WP 3.5.