I've had a similar issue surface recently. Core2Duo rig wouldn't do anything. I changed RAM, CPU, PSU, unplugged everything else trying to get it to show any signs of life. Even replaced the power switch, as I have had that happen once before. I had to leave and just left it there with power attached. I usually turn off the power strip, but forgot to as I lost track of time and had to go. Came back and it had booted by itself. No problems for a week, then did it again. This time I left to get a cup of coffee and came back and saw it booted. I have had a couple old motherboards refuse to cooperate with me and after trying everything, simply leaving it set for a while it came back to life.
Since I changed out EVERYTHING else, I'm guessing somehing on the motherboard is dying. Considering the abuse it has been through (running in a shed in Florida for the past 3 years) and the low price I paid for it, I really can't complain. You can try testing parts by swapping in a known good part if you have one available. Some old mobo's run flawlessly for over a decade and others just get flaky with age. If you have been overclocking, you may want to try pulling power and removing the battery and let it sit for a while. It doesn't make any sense but it has worked for me more than a handful of times.