Acer travelmate P215-52G Refusing to power on sometimes

On your TravelMate P215-52G the power button is an integral part of the keyboard, with the wiring part of the keyboard ribbon cable. In order to power on the motherboard you need to get a good contact between the two wires on that cable that feed from the power button. I'm not positive, but I assume the physical connection uses the keyboard membranes just as the rest of the keys do. So, anything that gets in to contaminate the spaces between the membranes could cause symptoms like yours, just like they could cause intermittent key strokes. You could also get the symptoms if there's something wrong with the wiring in the cable itself, but that usually just means it doesn't work, not intermittent. It could also be something on the connection of that cable to the motherboard, corrosion would typically present as intermittent until it gets bad enough to just fail. Here's a diagram showing where the keyboard cable connects to the motherboard.

Peel the tape holding it in place and inspect the cable for any damage. Re-seat the connector a few times to wipe any corrosion off. I don't have access to a pinout of that cable so I can't suggest any good way to test the function of the power button itself... As with most modern laptops, in order to keep them thin, the keyboard is integrated into the top case so just replacing the keyboard isn't a cost effective solution.