Hello Sean-John-Michael-and-Miriam-- I've continued to experiment since my last post, and (by following the advice of kind Miriam)... as my computer has continued to have bouts of not responding to the power-button at all, (intermittently), I've started the following process that then allows my computer to boot every time I encounter this problem:
1. I start by unplugging all the peripherals (in my case that's a Lacie superdrive plugged into a firewire port, a 4-port USB hub, [with just the plug-adaptors attached for my ipod, & cameras], and a Canon Pixma IP4300 printer).
2. Then I unplug the main power cord from my IMac G5, and re-plug it back in.
3. Then I hit the power-button and it starts every time!-- After it's started I plug back in all my peripherals... Go figure!?? Guess there's some type of electrical-glitch, (or software incompatibility?) between one of the peripherals and the "start up" functions of the Imac? Seems strange to me, but I'm glad it works consistently! Good luck John solving your non-start-up mystery-- hope it's as simple as going through this process.