Have you tried testing the other 4 cards from the other rig in this one?
I once had a card that when it failed it would 'lose' its voltage regulation and just pull more and more power at the wall until the PSU was overloaded and it would shut down. I witnessed it with a killawatt or I wouldnt have believed it.
I've also had cards that would do this to a machine if they were clocked a bit too high.
worth testing anyway..
I've tried every combination of 3, and even swapped cards between rigs. It happens when any 4 are installed. So it's not any one of the cards causing it.
The cards are overclocked, but not too high. I haven't tried running at stock yet, so that is worth a try. I'll try all 4 at stock and see what happens.
Edit: Remember that the problem did follow the power supply when I swapped that between rigs. So it feels like a power supply issue. But it's very odd that I got two that both didn't work on this rig, and one that does.