'incredible' is right! You need to check the largest files... if they are core dumps or log files (thats what populat /var most of the times), just compress the older ones.

df -h (use -k for Solaris 8 and below) will give you the consumed capacity of your file systems. Your company must be keeping some sort of threshold levels to page/call the on call person to address the issue. We are using 80% for major alerts and 90% for critical alerts!

If it is some other file system than /var and the owner of the culprit files is not root, contact your help desk to contact the file owner to address the issue.

Cheers!