In the new world of vm's and undersized disks, this will probably be an issue.
First make sure you are not within the margin of reserved disk space for the superuser (if it was created that way).
Usual defaults are around 5%.
Next, check your inodes. No inodes, no writes will happen.
df -i - to check where the biggest usage is: find / -xdev -printf '%h\n' | sort | uniq -c | sort -k 1 -n
- attribution of the find command above goes to: http://unix.stackexchange.com/users/4358/patrick
You might also want to check for deleted files that processes have open.
You might have a few big ones hanging around.