Well, I remember back in July 2003 I was called in on an 'emergency' status job (I'm a sysadmin). The /var directory was 97%. In this case, it had happened because there was an antivirus program on the server which was storing all emails that had been detected with viruses in the /var directory. The hosting company had aquired itself a virus spammer, and therefore totally clogged the /var directory. Simple solution for this situation was to remove the spammer and set the antivirus program to clear itself out every 6 hours.
There are of course many other causes of this sort of problem - large log files, etc.