I've been stuck on numerous rides at Disney and elsewhere but the most recent was back in the beginning of May when we got stuck on Big Thunder Mountain Railroad... It was my mother, my father, my boyfriend, myself, and my 4 1/2 year old son. We had just left the loading station and the first couple of cars had just made it onto the first lift hill when the ride stopped dead in its tracks. At first they kept broadcasting the "the ride will start up again shortly" messages (don't remember the exact ones though) then after about 5 minutes they turned all the interior lights on and then about 5 minutes later a CM came to talk to us and told us that they were trying to get it going again and that they weren't going to evacuate us because they should be able to get us going again in a minute. Then he left and about 5 minutes after that we had another person come back with him and they kept telling us the same thing. Finally after about 20 minutes or so total they realized that the ride wasn't going to be started up again any time soon and they had to evacuate us off the ride. We had to climb out of the side of the train (and the seats are higher off the ground that I realized) and jump down onto the uneven area next to the track and then try and walk single file on the rocky edge all the way back to the station. I was just glad we weren't one of the trains that was out in the middle of the ride when it happened.
The bad news was that they didn't help anyone to get out and my mom ended up falling and hurting her shoulder (the CM that was evacuating us really needs to learn about customer service - after my mom fell she didn't bother to help her up, or ask how she was even... and when my mom complained that her whole rear end was wet from the puddle on the ground the CM told her to go ride Splash Mountain and then she would fit right in). And to top it all off - they didn't try to help my son or offer us anything like free fast passes to use later or anything.