Tim, your comment about timelines put two things together in my brain that I looked at recently.
Last night, the automated course backups started at 12:50AM. It proceeded rapidly through 8 or so courses finishing them by 12:53AM. My course started backing up at 12:53 and failed at 12:56:01. It then proceeded through the rest of the courses and finished by 12:58. Now its pretty typical that my course takes the longest since I have an absurd number of questions and quizzes.
The part that is interesting here is that "Delete stale temp files" task ran last night at 12:56. In fact, all 4 of the automated backups that have failed in the last week all failed at either 12:56 or 12:57. This task is scheduled to start on the 55th minute every 6 hours.
So I should just be able to edit the schedule of the automated backups and delete stale temp files tasks and avoid the collision. With that said, I won't do that for now so that I can help with debugging. I'll see if I can track down where/if cron is being logged on our system. I have a feeling its going to /dev/null though.