The problem here is that /tmp, which (by default) is used for storage of session information, is full. A cron (scheduled task) should run periodically to clear expired files out making space for more. I'm not sure how often that runs.
If you wait the problem should eventually clear itself. The problem used to occur frequently at one time though instances seem less frequent of late. Nevertheless it could reoccur.
The alternative is to set up your own session save directory in your own webspace. A simple means to do this is described in Gabe's post
here. When the script runs it lists directories where links to your own php.ini have been added - this is normal.
David
