I once had a similar problem with a non-LVM partition that I solved by moving one of the directories to a more spacious partition and symlinking it back into place. In your case, for example, you can try:
mv /var/cache /more/spacious/partition/cache
ln -s /more/spacious/partition/cache /var/cache
Please note that I did this with a non-system directory and have so far observed no ill side effects. The case may be different with system directories, though. We need someone more knowledgeable to confirm/refute.
To be safer, you can do
cp -a /var/cache /new/place/cache
rm -rf /var/cache
ln -s /new/place/cache /var/cache
This ensures you won't lose your cache in case the
mv call gets botched somehow (system crash, power outage,...)
To ensure that nothing is being written to the directory while you copy it, it's better if you do this via LiveCD.