Accord LX

• Automatic transmission
• 125,000 miles

Periodically, while driving my car, the entire system shuts off (I have never had this problem trying to start the car. For the most part, I would coast to the side of the road, put it in park, and take the key out of the ignition. While it would usually turn back on after a few minutes, this last time it resisted all efforts until I waited about 10 minutes, revved the engine past 4000rpm while trying the ignition.
The car has not been trouble free; I have had to replace a cracked engine, several radiator hoses and the battery since I got it a year and a half ago. However, I have gone to two different mechanics to get it fixed, and neither has figured out what is causing this.
Initially, I thought it was in the electrical system, since the first time it happened I had air conditioning, lights, radio, and wipers on (it was raining), and I assumed I had overloaded it/had an old battery. After replacing it, it happened again several times. With a new engine, I am pretty sure that is not the source of the problem, but the fact that the revving caused it to work, I am now guessing that there is some issue getting fuel to the engine maybe.
Any thoughts would be appreciated, since no mechanic will ever say "I don't know" without doing at least $300 worth of various labor first.