On NUMB3RS last Friday, Peter MacNicol's character, Larry, who is the brilliant scientist who knows a lot about everything, made the very pedestrian mistake of saying "begs the question" when he meant "raises the question."
Far worse, though, was on the series finale of The West Wing. President Bartlet, an extremely knowledgable student of history, is asked by his wife whose idea it was to have Inauguration Day in January. Bartlet dryly quipped that she could blame Adams, Jefferson, and Franklin, implying she should stop complaining, unless she wanted to impugn our wise Founding Fathers.
Except, of course, that until the 20th Amendment, ratified in 1933, Inauguration Day was usually held in March. President Bartlet never would have made such an obvious mistake. And it's frankly quite sad that no one else caught the error before the episode was completed.