Here is a procedure to check things:
1. Turn off the circuit breaker.
2. Slowly run all of the water out of the tank while watching the water pressure gauge. At some point, the pressure will suddenly drop to zero. The pressure where it suddenly drops is the air pressure in the tank.
3. If you have a tire gauge, check the air pressure in the tank.
4. If the air pressure is low, add air to the tank until the pressure is about 3 psi LESS than where you want your pressure switch to start. With a water valve open, that should force all of the water out of the tank. If the pressure is too high, bleed the air down to 3 psi less than you want your pump to start.
5. Try to "rock" the tank to make sure it is empty of water. If it has a lot of water after step 4 then the bladder has probably failed and you need a new tank.
6. Close all water valves and run the pump for just a few seconds. The pressure should increase immediately to the air pressure, and then more slowly. Shut the pump off with the breaker and watch the gauge. It should hold steady. If it drops you have a leak, or an open valve, or a check valve failure.
7. Runs some water slowly until the pressure has dropped to about 3 psi above the air pressure in the tank (from step 4). Use a flat screwdriver to lift the pressure plate near the base of the switch to turn the switch to the OFF position if it in not already off, and then release the pressure plate.
8. If the switch stays OFF after step 7, then adjust the main screw/nut on the pressure switch to increase the pressure setting until its switch goes to ON. If the switch goes to ON when you release the pressure plate, then adjust the pressure switch to decrease the switch point until the it just operates when you release the plate.
9. Run the pump and observe the pressure where it goes off. Use the secondary screw/nut to adjust the OFF point of the pressure switch without changing the ON point. There will be a minimum ON/OFF differential that you can achieve of 15 to 20 psi.
I had a tank where the failure was not the bladder but the tank on the air side of the bladder. If your tank loses air pressure, then you must repalce the tank.