Don't think that is the result of a bad checkvalve...
There is a pressure switch that (normally, anyways) turns the pump on and off. This could be shot, or the pipe going to it could be plugged so it can't sense the pressure changes. How do you turn it on manually? Some pressure switches have a low-pressure cutout - if the pressure gets too low (like where the pump is pumping air because the well water level got too low), it requires a manual reset.
If this is the case, it could be you litereally are running out of water, or the bladder tank's pressure is set incorrectly (too high). It should be set 2# less than the turn-on pressure setting. You can test this with an air pressure gauge - shut the pump off, open a faucet, let the water run until it stops, then check the pressure on the tank. If water comes out it is shot. If not, adjust it to the proper pressure and see what happens.