Mac Pro :: Getting A Sporadic Lock Up Where The Screen Freezes?

I have a first gen Mac pro, I am getting a sporadic lock up where the screen freezes, and it stops responding to keyboard and mouse input (and it becomes inaccessible from the network). This has happened in normal use twice.

I have tried several times to run the Apple Hardware Test off the original Os X disk, but it locks up up every time in that mode. It also locks up immediately when I boot into single user mode (which I did in order to attempt to run memtest). At least it refuses keyboard input in Single User (with my normal bluetooth keyboard, as well as a USB keyboard that I borrowed).

I did run memtest from a terminal window while logged in normally, and it was able to access 11 of my 12 gigs of RAM, and it passed, so I guess I am 11/12ths sure it is not bad RAM.

When the system does lock up, when I do a hard power cycle with the button on the front of the machine, it will not reboot -- my screen remains black. If I let it sit for a minute or two like that, then power cycle again, it does boot up normally. This is very consistent on these lockups - no boot the first time, works fine the second.

In general, I think it is running hotter than it used to -- at least it seems like the fans are running at high speed more than they used to, but I can't swear to, or quantify that.

I was on the phone with Apple tech support for about an hour yesterday, and he finally decided it had to be taken in for service.

I wanted to see if anybody here has any ideas what the problem might be, especially if it is something I could fix myself, before I lose the machine for however long it will take Apple to fix it.