A problem I have never really found a satisfactory answer to. On some computers in some places, I have found that occasionally one has to disconnect them from mains to get them to power up occasionally. Some people have said I needed to discharge some capacitors but that never felt satisfactory. Very heisenbuggy.

Like Tom Knight, I was able to fix a broken machine by slowly power-cycling it :)

I'm not sure how the discussion went from art to power supplies, but... Turning off a computer doesn't completely turn it off. Specifically, ATX power supplies have a standby +5V output, even when the computer is off. (This is how Wake-on-LAN works, for instance.) So it's quite possible that you needed to disconnect the computer to get it to fully restart.

As far as discharging capacitors, that's an issue if you turn the computer off and on rapidly; the power supply can continue supplying power for a few seconds due to its capacitors, so the system might continue uninterrupted.

I was a naughty boy and would occasionally try to cycle the switch fast enough that the machine didn't turn off.

This reminds me of power-glitching my Atari 2600 (or "frying", in the 2600-enthusiast vernacular) to get odd "effects" in some games. It randomized the state of the system and, given the relative simplicity of the platform and the "one big loop" nature of most games you could actually get some interesting and useful results.