Four minutes before the moon landing, the Apollo Guidance Computer began throwing 1202 and 1201 alarms - indicating that it wasn't completing its processing loop in time.
Here's the audio from the flight director and guidance loops, as they troubleshooted the problem in real time, clearing Apollo 11 to land.
tl;dr: Some misconfigured switches meant that rendezvous radar jobs kept being erroneously scheduled, which used up available memory. Whenever there was no more memory available, the alarm would fire and the computer was rebooted. It would immediately restart (resuming from near where they left off) all the high priority jobs (while not restarting the radar jobs), meaning that it could still land safely.
The AGC was one of the first to use priority multitasking. The 1201 and 1201 faults were indication that lower priority tasks were getting yanked.
See "Moon Machines". Seriously.
A couple of links to the videos you mentioned:
http://www.sciencechannel.com/video-topics/space-videos/moon...
The above is probably one of the most informative 2 minute videos I've ever seen.
https://www.youtube.com/watch?v=DWcITjqZtpU&list=PL7E6922816...
"there were no specs, we made it up" "if you screwed up you got a stack of papers that was 2 feet thick" "the overall memory for the Apollo guidance computer was 72 kilobytes"
Just another kind of "the space program was amazing" thing - the sheer diversity of people who got hired on to contribute is continually surprising. An amazing tribute to the pragmatism of NASA under the conditions.
The problem was actually a deployment problem. The checklist for configuring the computer left a diagnostic mode on that sucked up a lot of CPU.
We can make web sites that stay up, always. It's just that it costs a lot more and / or they do less.