Every issue that has been speculated on should throw a code on the display, if the logic board is working correctly. Inadequate flow or clogged coils will cause high head pressure and throw a HI-PS error on the display. Low coolant will cause the compressor to ice, low pressure and LO-PS display. AC voltage dipping would cause the unit to shut down with a LO-AC message.
I have a split gas system on my boat and just had one of the units fail due to what turned out to be a bad logic board. The failure manifested as the unit shutting down exactly as Doc describes, except mine would shut off after 1-15 seconds of the compressor running. The fix was replacing the logic board. Cost about $275 and was fixed in about half an hour. One word of caution, many of these units, mine included, are 240v. Make sure you turn off the power before cracking any of the covers, 240v shocks aren't quickly forgotten.
Ben