On one of our AIX systems (5.2) when we try to run smit we receive the
above error message and cannot proceed.

Doing a df shows plenty of room in all file system. Plenty of inodes
and plenty of blocks available.

Perhaps a related issue (also strange) is that on this box logging in
as root, trying any command such as ping or ls the error received was
"access not allowed". Following this we rebooted and this error
seemed to go away.

If anyone has received and overcome this error, please advise what
might be done to overcome it?

Thanks in advance.

Please check the profile of root,or you copy a profile of root from
other AIX system.
And try it again.
Just an advice.