At first I thought the water valve was open for too long and giving too much water.
After thawing out the ice maker and then putting it back, I found that the first batch of ice was fine. But the ice maker didn't dump all the ice, so the water (plus the undumped ice) for the second batch was too much. This continued until the whole ice tray got full of ice.
I was thinking that the ice tray had become a little too sticky so the surface was holding onto the cubes.
I chatted with somebody from AppliancePartsPros.com. She said that probably the dump motor was missing some teeth and wasn't twisting the tray as much as it should, and therefore I should buy a whole new ice maker assembly.
Worked! Hardest part was removing the food shelves from the freezer. Don't forget to watch the instructional video.