I tried resetting the 'tmp' directory (in case that partition was hokey or something). That didn't work. Then while surfing around the MySQL site, someone mentioned using the perror command. So I did that and it tells me "Disk quota exceeded".
So then I scrambled around to figure out quotas & how they worked on Linux. To find out a users' quota (you may need to be root for all this):
quota -u [userid]
Then to change the quota:
edquota -u [userid]
And voila - problem fixed!
I would like to disable quotas altogether, but I'm using a fancy Cobalt RaQ system and don't want to mess it up too much. Cobalt has a fancy admin website you can use to make those type of changes, BUT only for real userids, not system ids (such as 'mysql').