This trope was deliberately invoked for the PC puzzle game Sheep, where you play a herder driving a flock of sheep around various obstacles, and you had to compensate for the flock's tendency to go in the wrong directions and frequent suicidal stupidity. The game even lists this artificial stupidity as a feature.
The LEGO games are especially good (bad?) at this. Although it has improved in recent games, more than often your partner will be busy walking into a wall on the other side of the level, while you need him to come and use his ability.
Although it's understandable given the insane number of things you can work with, the AI in Scribblenauts will stick to its guns regardless of whether it makes sense or not. This leads to amusing situations like a group of witches accidentally transforming one another into frogs while fighting an enemy, and then the survivors eating their frog-ified sisters.
In Nuts & Milk, Nuts has a tendency to suicidally plunge into the water at the bottom of the screen, but makes up for it by respawning quickly.
Puyo Puyo!! 20th Anniversary 's Core AI might seem powerful at first, but it can sometimes place its puyo at the worst places possible and even kill itself, see for yourself here.
The developer of "The 11th Hour" stated that trying to add the correct amount of Artificial Stupidity to the AI was one of the major factors in the late release of the game. This because an issue because in its prequel, "The 7th Guest", Stauf's AI is so strong that many players abandoned the entire game due to inability to pass the puzzle where you play against him.