Though I'm sure you've already done so, make sure your switching power supply isn't set too low. Most are set at exactly +5v. Pac-Man boards expect up to +7v input as the board itself down-converts the incoming voltage to the +5 it needs. I generally crank up the adjustment knob on switching power supplies going into Pac cabs to as close to +6v as it will go to ensure it never falls below what is expected. If your switcher is outputting anything below +5 it will cause intermittent issues.
I would almost wager a bet on the above being your solution. When you power on your other games it is probably draining supplied power to your circuit just enough to cause problems.
It sounds like your wall receptacle is straining with too much plugged into it.
You could also correct the issue by adding a second breaker to your breaker box and dedicated lines specifically for your arcade games, but if you aren't experienced in doing so, that can be costly.
Try the above and see if it helps...if not, post a few pics or a quick vid so we can get an idea of what you are seeing.