Here's an example of a bad brain teaser:
There's a room near you with a light in it. You're in a room with three light switches. One of those switches controls the light in the other room. You can do what you want in this room, and when you're done, you can walk into the room with the light. Then you need to say which switch controls the light.The supposed answer is: leave one switch on for a while, then turn it off, and turn on one of the other switches, then walk into the room with the light. If the light is on, it must be the most recent switch you turned on. If it is off, feel the bulb. If the bulb is hot, it must be the first switch, otherwise it's the last switch.
However, this supposed answer assumes some things that were not clear from the problem statement. First, it assumes that the light in use is the sort that heats up. Second, it assumes that the light is within reach, and not attached to a high ceiling. These assumptions are not clearly stated, so someone thinking about the brain teaser can't assume they act in their favor.
Another example that bother's me a bit is the Candle Problem (my phrasing is taken from wikipedia):
How do you fix a candle on a wall (a cork board) in a way so the candle wax won't drip onto the table below using only the following (in addition to the candle): a book of matches, and a box of thumbtacks.The supposed solution is to empty the thumbtacks out of the box, put the candle inside it, tack the box to the wall, and then light it. However, I'm not convinced that it is clear to all participants that the box holding the thumbtacks is a legal item. It seems fair to think that the experimenter simply included the box so the thumbtacks wouldn't roll away. If it's fair to use this box, why isn't it fair to move the table away from the wall?