Pressure jumps fast between 35 to 50 PSI

With water running from the faucet, my well pump cuts-in around 28 psi. When it slowly reaches around 37 psi it rapidly jumps to 50 psi and then the pumps cuts-out. Then the pressure drops rapidly down to around 37 psi. From 37 psi down to 28 the pressure drops normal. What is causing the rapid pressure increase and decrease between 37 and 50 psi?

I drained the water in system until 0 psi on gage. I checked tank pressure with an air gage. I could not add more air above 21 psi using a bicycle pump.

Why could'nt I add more air to the tank?

You may have a tank leak on the air side of the bladder. It doesn't take much leak to be large enough to leak out at fast as you can pump with a bicycle pump when you get to 21 psi.

After you pump it up, immediately check the air pressure and then watch it for a few minutes. If you lose air in a short time, then the tank is leaking.

I chased a problem like that for a while and discovered the air leak at the welded fitting near the outlet on the air side of the bladder. It was not possible to see it when the tank was installed.

You might be able to check it with a little water in the tank. When there is water in the tank the water pressure gauge will read the air pressure. If the pressure drops with no water usage and pump off, that is another indication of an air leak in the tank.
Bob NH

Most likely the bladder tank needs to be replaced.

You didn't describe the tank. Could it be a WX-203 Well-X-Trol?

You are topping out the bladder. That's why it jumps from one pressure to another. The bag has hit the top of it's travel. Some 203's had a dome in them to prevent tearing the bladder from overextension. When the bladder reached the dome, the tank could take no more water so the pressure rises immediately to shut off.

bob...