Transient startup effects can make a difference. With that GPU that I was talking about, sometimes it was fine for a long time, sometimes it worked for a little while and sometimes it erred on POST or the windows splash screen. I know it was the GPU, but boy it was not consistently reproducible.
AMD CPU's run hot and I would suspect it (only suspect), but if the fan caused a power drain then you can't be certain. Since you probably have SDR memory then you will have to make some major choices about your repair/upgrade.
http://www.computergeeks.com usually has a good selection of older motherboards for cheap to allow you to continue to use your old memory. Their tech support has gotten sort of crappy but I am hanging with them as they used to one of the best in this regard and I am hoping that they will return to being so.
I wouldn't even rule out the power supply, if you upgrade to more modern technology you will probably need to upgrade it anyway.
The fact that the memory test ran fine also seems to say that the processor and some elements of the MBD were ok for that long at least. You might try disconnecting some of the attached devices and see if this clears up the problem. If so it is either these devices or the PS. Keep in mind that with an intermittent error, even the most rigorous tests may be passed. I would think about repeating the test if the problem continues/gets worse.
If you do upgrade, feel free to PM me for ideas about what and where to buy. I suspect that you are knowledgeable about these things yourself, but I do this sort of thing all the time and if you wish another opinion feel free.
Best.