Any ideas why the ice maker in Maytag fridge won't work properly?

this is a 2007 no frills freezer-on-top model that came with the home we just purchased. The "full" lever pops up on its own, shutting down production of ice. It has never produced more than maybe 3 cycles' worth at a time without just shutting down. No online resources have provided any troubleshooting advice. I pulled the 'fridge out and the water feed looks fine. Beyond that, I'm at a loss. any suggestions? Thanks