If the file system is full, users cannot login via gpm: the system tries to start an X session, but fails and returns to the login window. The user is not informed what causes the failure.
The problem here is gdm lacks any facility to display errors or dialogs of any kind, it only logs them to syslog. It's a large fix that's required.
I looked into fixing this more; the problem is actually worse than I thought. gdm can display errors, up until it exec's your session. After that it's hard to know what to do; gdm isn't running out of space I don't think, I think the apps launched in .Xclients are running out of space. At that point gdm is out of the picture. So, this bug is actually "many apps have bad error handling" I don't think it's a gdm bug.
This problem is basically unfixable since it's so general; specific bugs against specific apps would be required.