This machine had a stale NFS mount. But attempting to unmount gave a rather strange message:

$ umount /mnt/nfs
/mnt/nfs was not found in /proc/mounts

Let's look at mtab first:

$ mount | grep nfs
server:/mnt/export/dir0 on /mnt/nfs type nfs (ro,vers=4,addr=10.0.0.10,clientaddr=10.0.0.30)

OK, but why isn't it found in /proc/mounts then?

$ grep nfs /proc/mounts 
server:/mnt/export/dir0 /mnt/nfs\040(deleted) \
                   nfs4  ro,nosuid,nodev,noexec,relatime,\
                   vers=4,rsize=131072,wsize=131072,namlen=255,\
                   hard,proto=tcp,port=0,timeo=600,retrans=2,\
                   sec=sys,local_lock=none,clientaddr=10.0.0.30,\
                   minorversion=0,addr=10.0.0.10 0 0

Hah! Somehow the entry in /proc/mounts got mangled. OK, \040 is octal for SPACE. But umount(8) still fails:

$ umount /mnt/nfs\ \(deleted\)
umount.nfs: /mnt/nfs (deleted): not found
umount.nfs: /mnt/nfs (deleted): not found

Apparently umount(8) tried to find the entries from /proc/mounts in /etc/mtab but failed, due to the mismatch of both files. After adding \040(deleted) to the local-directory part in /etc/mtab, umount(8) did something:

$ umount /mnt/nfs*
umount: /mnt/nfs Stale NFS file handle

The entry got removed from /etc/mtab, but the mangled entry still shows up in /proc/mounts. Now all is left is a stale NFS file handle:

$ umount -f /mnt/nfs
umount2: Stale NFS file handle
umount: /mnt/nfs: Stale NFS file handle

Sadly, this could only be cured by a reboot :-\

Update: the "Stale NFS file handle" error is covered quite nicely in this bugreport. According to this report, the error should be a thing of the past since Linux 3.12.