I've seen A600s which develop a problem where they won't boot up after being switched on, but if left for 5-30 minutes they will eventually boot. It's caused by bad capacitors, which nearly all A1200s, A600s and CD32s suffer from these days.
Try turning them on and just leaving them for half an hour, see if anything changes. I'd advise you to change all the capacitors as soon as possible if it seems to be the same problem I've encountered before.
