Time Machine couldn't complete backup
When Time Machine begins a backup, it first 'prepares' the backup. It's at this point that the space requirements are calculated. If there's not enough space, it will delete old backups, and tell you so, including the date of the last backup that's left on the drive.

However, your error is that it's run out of space unexpectedly. This is much more rare.


Try repairing your drive with Disk Utility

There might be a problem with the disk and/or partition, where directories are smaller than they might first appear. Try repairing it.
Another process is writing to the drive

Make sure that nothing else is being saved to the partition at the same time. If another process writes a big file to the partition, then the space requirements won't match, and you'll get this error.