In college, I was frequently invited to movie nights in one of the lounges because the TV there was ancient and I was one of the few people who could reliably turn it on. It involved hitting the reset button at a specific point of the warm up for a specific duration. Apparently no one thought to just ask for a new TV until years later.
I had a computer in the pentium 3 days that eventually would not turn on while upright. Laying the case flat (as you would when opening it up to root around looking for the problem) would allow it to turn on again. Once on it could be set upright and keep running till next boot up.
This was only discovered after several months of intermittent frustrating debugging.
I can see why you'd think of it as a version of this story, but it's also a version of "Tom Knight and the Lisp Machine" (http://www.catb.org/~esr/jargon/html/koans.html#id3141171).
One day my dad was trying to reboot it and it wasn't cooperating. He'd flip the power switch to off, wait a bit, then back to on, and the fans would spin up but nothing else would happen. After seeing this repeat a few times, I gave it a shot -- power switch off, wait a bit, power switch on. It booted up happily.
The trick was that the power supply took a while to fully shut down after losing mains power. If it regained power too early, it wouldn't start back up again. The only people in the house who could reliably reboot that machine had ears young enough to hear the high-pitched whine of the capacitors, to judge whether the reboot had finished and power could be restored.