When I start a game the screen turns black and the Wii freezes, pulling the power cable is the only way reset it (holding the power button does nothing). It happens not only on Wii games, but also on gamecube and even virtual console games.
edit* just thought I'd mention that the Wii is sitting out in the open, so its not from overheating inside of a closed cabinet. The first time I found that it was broken was after about a month of not playing any games on it, but Connect24 was on the whole time and some online theories are that that is the cause of many of the problems (since the Wii is still generating some heat, but has no fans on).