Thank you Simon for the fast response!

>>That the folder is named .Trash-0 suggests that the files were deleted
>>by root since the 0 was the UID of the user (root is UID 0).

This is correct

>>How did you delete the backups and log files? Perhaps through the GNOME
>>(GUI) desktop using Nautilus?

I did use the default Browser which I just learned is Nautilus 2.28.4
I right clicked the files, and selected Move to trash.

>>If so, by default that acts like Windows' Recycle Bin.

I agree, except that when I look in the trash can that is on the desktop there is Nothing there.
If I click View, select show hidden files, I can browse to /var and I do see .Trash-0 folder.
My concern is that I cannot empty the trash and "everything" I delete stays there.
This is the issue I had and my monitoring software (Nagios) was alerting me that /var was 87% full.
How can I "Actually" delete something if it doesn't appear in the trash, (but does in the hidden .Trash-0) ?