Regardless of whether you've changed permissions or not. Check them.
You may need to do a permissions repair on that disk in Disk Utility.
Thank you for the quick reply. It didn't work though.
I looked in Finder at the folder where my backups were being stored and it still says "read and write" under privileges.
And I found the disk utility and the button that said "repair permissions" and clicked on that. It took a few minutes and showed many results about other things but not Lightroom. I copied it and saved it in case that would be useful. Let me know; I won't post it now because it's long.
Unfortunately, even after restarting my computer, Lightroom is still failing to back up, giving the same error message as I posted above.
Anything else I can do?
Would a screen shot of the folder where my Lightroom backups are stored help? Next to the Backups folder, which I created 2 years ago, I notice another folder called "Corrupt Catalogs" which I don't think I made myself. According to file info, it was created December 18, 2013. It has 5 catalogs in it, the latest dated July 13, 2014. I have no idea if this is related to my backup problem; just mentioning it just in case.
As for the Backup folder, it has 3 backups in it dated 9/30, 10/02, and 10/05.
I'm still hoping to get an answer on this. This problem persists.