Simon,
What do you mean "I checked the power supply." The
only way you can adequately check a power supply is to
connect a voltage and frequency analyzer to both the AC
and (voltage only) to the DC output(s) and record the
values over a long period of time. You could have
momentary changes that you will never see otherwise. I
am not necessarily suggesting that power is your
problem, but it could be. Are you connected to a good
UPS, and does it ever indicate it is switching other
than during storms or when you would otherwise expect?
That would be the next best thing to a power analyzer,
which could cost you more than it is worth.
If you suspect hardware, then carefully reseat all
the boards and connections. Since you don't get an
error message on startup, it is probably not a loss of
power that actually shuts down the computer, but rather
just a momentary signal that is seen only by the
shutdown function. Could be as simple as a little
corrosion on the terminals of one board, or a
particular component that happens to be heat sensitive.
I think replacing (or just swapping position of) the
memory chips is probably a good idea, since they are
the most likely to be selectively sensitive. There is
some memory on the CPU board that might also cause
weird problems.
Good Luck.