You cant really trust some digital multi-meters in a case like this. They draw such a tiny amount of current that they will measure voltage created by capacitance or inductance, which is in the milli or even micro amps. Usually putting any kind of load at all on the circuit will drop the voltage to zero.
At least thats what I was thinking until you said the lights on the plug-in tester glowed very dimly.
My first thought would be that the circuit in question is getting "bleed over" from another circuit, could be two (un-insulated part) wires barely touching inside a junction box or the breaker panel.
Also, have you measured voltage hot to neutral, ground to neutral, and hot to ground?
In a USA type house wiring, the hot goes back to the breaker, or fuse, and the ground and the neutral tie together in the circuit breaker panel, or fuse box.
So with the breaker on, you should read ~110VAC hot to neutral, hot to ground, but nothing ground to neutral.
One thing you can do is put the volt meter on your suspect circuit, then turn off one breaker at a time to see if this circuit is somehow getting some bleed over from another circuit.
If the voltage goes to zero when you turn off a certain breaker, then that is your problem circuit.
Fixing this problem may not be so simple. But I would start by checking the wires in the ceiling lights, these usually are junction boxes too, however I dont run more than one feeder circuit to a ceiling box, but I have seen it done with 2 feeders inside the box..
Just to verify that the problem is not in your cab wiring, have you measured the voltage on the wall outlets with your arcade cab unplugged?