Never give up!
I'm just throwing this in for ideas but it's very unlikely to be the case in yours.
I had a killswitch relay with my alarm which I needed to wire up to the coil, I'm sure it was wired up properly, but it cut out a couple of times at low speed. (scary stuff when the power steering cuts out with it)
Luckily I always keep a few tools in the car so disconnected it and crimped the wires back together, it's been running fine since. Looking through other cut-out problems, it could be all sorts of things, but think logically. The engine needs fuel, air, a spark and to be moving to get the cams and sparks working.
Rule out the last one and you're left with air, fuel and electricity.
Air is the easy one to check so unlikely.
My vote is on the electrical system as above. It sounds like a loose connection or damaged wire. Like my problem, it disconnected and reconnected on vibration so if you can, run at idle and shake it. (bouncing it probably won't be enough)
Does the stereo switch off as well?