As Doc C mentions in this thread,
http://eiiiforum.com/index.php?topic=1211.0, I think the ribbon cable was getting pinched. I pulled all of the excess cable towards the back of the machine and the problem has gone away. (I also made sure the cable was threaded through the board chassis which it was.)
I have encountered this problem before, and in my opinion, misplacement of that damn cable is the easiest mistake a repair novice such as myself can make in regards to the EIII. I wish E-mu had come up with a hinged top and a better solution than the ribbon cable. Maybe there is a modern alternative?