Panasonic Model NN6381A
I can always enter the power and the time settings.
When I press Start, one of these three things happens:
1. The oven does not start and 88:88 appears in the dispplay
2. The oven runs for a few seconds, and then stops and 88:88 appears
in the display
3. The oven runs properly

What might cause this? Behavior #2 above makes me doubt that the
keypad is the problem.

Try resetting the oven's SYSCON.

Unplug and leave it that way for about 10-30 minutes. Plug it back in, set the
clock, and then use it again. - Reinhart