No space left on device -- but there is space

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http://serverfault.com – I'm running an Ubuntu 12.04.1 server in a VMware virtualized environment that has a strange issue. Twice now, at seemingly random times, the machine has suddenly been unable to write to the root partition despite plenty of space seemingly available. I'm at a loss trying to figure out what's wrong. "df -h" shows all mounted volumes with 40% or more free space. Both times this has happened, rebooting resolved it for a few minutes, and then it was back to saying "no space left on device." Restoring it from a snapshot seems to fix it for longer, but since this has now happened again, I'm not sure (HowTos)