Please try one of the Joomla templates and see if the problem persists.
If not - then its your template that is causing the problem.

Just want to share an experience i just had with an install where saving users didn't work:

in the backend we got an error like this when saving a user:

[code:1]DB function failed with error number 1054
Unknown column 'sbsignature' in 'field list' SQL=INSERT INTO jos_comprofiler ( `cb_.......
[/code:1]

if you use JoomlaBoard and CB with integration, and this happens, you need to remove these fields and create them through JoomlaBoard's configuration pannel (there is button there to do this). The "cb_" prefix when you create fields manually for JoomlaBoard are not usable for JoomlaBoard.

Maybe one or the other has this problem...

How it can happen ?

Well if you interrupt things in backend on a overloaded server by clicking another link before an action is finished will interrupt the previous action by default.

Also MySQL might return errors on overloaded servers.

Anyway just try avoiding overloaded servers...