Will only start up some of the time, won't sleep

Ok, so here's the problem:

When I first try to power it on, the computer seems basically non-responsive--it shows no outward indication that the power button was even pressed. But, anywhere from 1 minute to 1 hour later, it will try to start up. I can hear the power coming on, but then the sound will stop abruptly, and this will occur several times in rapid succession, after which the computer will sometimes turn on, but other times (and more frequently) will just stop trying completely; then, the whole process will begin anew anywhere from 1 minute to 1 hour later.

Eventually, this entirely chain of events triggered by just the one press of the power button will successfully start up the computer, and then it will work just fine. It continues to function without a problem indefinitely, as long as it doesn't run out of power or attempt to sleep. It can run on the battery or the power adapter, and the battery charges and retains its charge. If the computer runs out of power, however, or if the screen is closed (i.e. if i try to put it into sleep mode), it will not turn back on, and I have to press the power button and hope for it to turn back on again whenever it decides to do so.

I've reset the SMC and PRAM, changed the battery, opened up the computer (no visible signs of water damage) and removed the optical drive (thinking that might have somehow been messing with things electrically), and replaced the hard drive--none of it worked. However, when I put in a different hard drive, it seemed to attenuate the problem; the power button still would not immediately start up the computer, but it would start on its own a few minutes later, and it would do so without the 'attempted-power-on' stuttering.

I'm at a loss for what to do. Might it be an unfixable logic board problem? Should I reinstall the OS? I don't see how that would help, but maybe I'm wrong?

Please help!