Amusing. Maybe somebody is sneaking into your house and zapping it with a Taser every Friday. I still think it's time to just hit it with a hammer and get another one, given all the time that has been spent troubleshooting it.
RE: Lavarock's theories:
a) Microsoft's patch release schedule is the second Tuesday of every month, unless there is an urgent security incident.
b) The whole point of a (properly functioning, good quality) UPS is to make power interruptions almost invisible (or at least insignificant) to the load. At most, the test would cause a few milliseconds of loss, as long as the battery isn't dying or dead. Unless this is a crappy UPS with a faulty battery, and too many things plugged into it, I'd doubt that is the issue.
Since the OBi was failing when NOT on the UPS, I think it is safe to say that it's just gone bad.