IBM AIX: cannot log in because its not the lowest login shell

Sometimes, common problems lead to very strange error messages.
Last time I got the following message when I tried to login via telnet to an IBM IntelliStation running AIX 5.1: You must "exec" login from the lowest login shell., and the login process failed.

The reason for this error is trivial: the root filesystem (/) was 100% full, so the solution was just to delete some unneded files (some old cores, in my case) and everything was fine.