I have a very old shallow well (only 30 feet deep) We just installed a submersible pump a few months ago. Sometimes we have water sometimes we don't. When the water comes back, we have mud in the water. The pump kicks on when it gets down to 30 psi and kicks off at 50 psi.

I'm wondering…is there a filter that could be getting clogged somewhere?

No filter, my guess is the well is too shallow and you are sucking up some bottom mud and periodically running the well dry.

Good guess Brian,
Sounds to me like a low water table. The pump kicks on and run until it used it up the water and then the mod filled in and the pump set it up. The solution would be to pull the pump and dig the well deeper. Good luck, Tom