Yup, I remember this. There were a few mini-games that involved the control stick rotating. Another brutal one was the single player game where you needed to pedal the cycle to light up the light bulb to keep the Boo away. I remember destroying my hand trying to get past the level in the single-player mini-game mode (that fun mode that was exclusive to the first Mario Party). Not only that, but it totally stripped off the paint on the area around the control stick. Fun times.

I never broke my stick, but yeah I'd heard there were many reports of broken joysticks (see above post).