What you have is a pressure switch with a low pressure dropout. The low pressure dropout is a safety feature, built in, that shuts the pump off should you rupture a line for instance.
Most of these are set to stop the pump when the pressure drops to about 20 psi. If your pressure switch is set to allow the pressure to drop that low before the pump cuts in, it will drop out.
Set your pressure switch to operate at 30/50, with the tank precharge set at 28 psi, and you'll not have any more trouble.