2 thoughts come to mind, one is memory, which can be given a full check with something like memtest86 from memtest.org. the bios test done during startup is not particulary detailed and can miss a failing address.
the second thought is mother board capacitors, dell had a problem with a machine gx270 series around 2007, anyways if the capacitors on the mother board are in anyway expanding, leaking, or going brown its the end of the road for the motherboard, I came across one recently that had a wide range of symptoms from failing to boot, moments of stability for up to half an hour, crashing just after loggin, rarely twice the same.
Rick