I had a similar issue with my Time 2000 where the pop bumpers (and probably the slingshots) would seem to start firing randomly after a few minutes of play. My problem ended up being the "Switch Common" line that is normally pulled up to 5 volts when inactive. This line was being pulled down by an issue with SW1 position 6. When this switch was in the "On" position, it was pulling the line voltage down to around 2.5 volts (right around the high/low voltage threshold). My theory is that when the chips in the circuit warmed up it would float further towards ground causing switches to be inaccurately sensed as closed.

My initial thought would be to check your "Switch Common" line to make sure that it's sitting at 5 volts when no switches are activated. Then one by one activate each switch and try and see if this triggers the problem. You should be able to monitor the voltage level of the common line with a DMM or Oscilloscope.