I've been a bit busy the last few days, but have been following along with this thread. You are getting good advise, so far, so really didn't feel the need to jump in until now. About the only thing I would add, is that I would also run a memtest86 test of the RAM on the machine that has been disconnecting.
Ram can do funny things, and cause all sorts of odd behaviors. If you have more than one RAM stick, and errors do show up. test the RAM sticks individually to see which is causing the problem. Sometimes all that is needed is to reseat the RAM sticks to make them behave. Other times they need contact cleaning. The MB RAM sockets can also become dirty, and show up as bad RAM, just because the contacts are no longer tight. Cleaning the sockets regains full contact and the problems go away.
The weirdest thing I have recently come across, was a 1GB stick of DDR RAM that reported errors no matter how clean I got the contacts, or what socket I plugged it into, in any machine it was tested in, so I labeled it bad, and laid it aside. Out of curiosity, I let it sit on my desktop for a week, then retested it. On retest no errors were reported in any socket. (Long overnight tests, in various machines, each night, for a week.) I reinstalled it into my backup computer, where it has been running flawlessly for the past six months. I can only conclude that something caused it to take on a charge that prevented proper paging, and that leaving it out of the system for a week allowed the charge to bleed off. I still consider this a test in progress, as I'd never heard of RAM being reported as bad, then recovering to full functionality before this. I find this rather interesting.