Classic II's have terrible capacitors... Whilst there may not be corrosion on the board, they could still have leaked and the goo could be shorting contact pads. Whilst a cursory glance may show the board as looking good, a closer look may reveal leaked electrolyte which may show as stickiness or a dry waxy film. One way to temporarily get it going may be to scrub firmly all the IC legs and pads and other electronic contact points with a toothbrush and see if the board becomes functional again. This has revived a few of my boards temporarily... this being said, if it does, you almost certainly have bad caps and the problems will be recurring and eventually the cap leakage will damage the logic board, sometimes irreparably if left too long. Even if the caps arent leaking  (i'd be surprised if some arent by now on a Classic, SE/30 or ClassicII logic board), they may still be open, shorted or otherwise out of spec and affecting the functionality of the asembly.

In short, I would thoroughly clean the board and then remove all the old electrolytic surfacemount caps ASAP before they leak anymore and have the board recapped.