If there is a foot valve at the bottom of the well, try taking out the checkvalve at the pump. You should not need two checkvalves. The one at the pump might be sticking shut.
Also, check the precharge on the bladder tank. If cut too fine (too much air) or watterlogged and there is no reserve left at the kick-in pressure, the pressure will drop to zero when the pump comes on. A jet pump uses pressure to make more and relies on a bit of reserve from the tank.
It could also be the pressure switch has crudded up under the diaphragm.