One example I can cite is the last Kaiser, Wilhelm II of Germany. He was not precisely stupid, but his family (including the other monarchs of Europe) and close advisers realized he was a braggart and unfit for leadership. Due to an injury at birth, one of his arms (I can't remember if it was left or right) was withered and useless. He went to great lengths to hide this weakness his whole life, and was very insecure about it, but naturally anyone close to him knew about it. As a result, he essentially always talked a big game about his martial knowledge and virtues. I wrote my final paper for my BA about him and his inner circle, and in many diplomatic records, there are notes about him making ludicrous remarks about military strategy, which his officers were quick to commend, but then ignore. This show essentially caused him to drive away many of the best leaders of the German empire and install yes men in their places. It was largely this insecurity and militarism the drove him to commit all of Germany's military resources to Austria in the wake of the assassination of Archduke Ferdinand. I argue that this was a major factor in turning a regional dispute into a world war.
To answer your question more directly, in the case of Wilhelm II, many people realized that he was basically an idiot, but since he had absolute power over the military, and a tremendous amount of power over the civil government, there was nothing anyone could really do to stop him from ruling. And partially as a result, we got WWI.