Troubleshooting well pump low pressure shutoff
I'd start with the easy basics - check the pre-charge pressure in the storage tank .vs. the set pressure on the well pump controller. I'm making the "generally good unless your well setup is very old-fashioned" assumption that you have a bladder or membrane type of tank. 

Before checking it, observe the pressure gauge as you run water, and note at what pressure the pump starts running (cuts-in) and stops running (cuts-out.)

Turn off the pump and drain the system. Check the air pressure in the tank - it should be about 2-3 PSI less than the pressure where the pump starts. If it's not, adjust it, so that it is. 

If the precharge is too close to the cut-in-pressure, you can run out of water before the pump starts building pressure, and that would cause a low-pressure cut-out, as the pressure will rapidly plummet once the tank is empty.