

The Last Samurai
The Japanese in the late 19th century did hire foreign advisers to modernize their army, but they were mostly French, not American. Ken Watanabe's character was based on the real Saigo Takamori who committed ritual suicide, or "seppuku," in defeat rather than in a volley of Gatling gun fire. Also, it's doubtful that a 40-something alcoholic Civil War vet, even one with great hair, would master the chopsticks much less the samurai sword.
This one movie has given entire Anthropology departments migraines. Sure the Maya did have the odd human sacrifice but not to Kulkulkan, the Sun God, and only high-ranking captives taken in battle were killed. The conquistadors arriving at the end of the film made for unlikely saviors: an estimated 90% of indigenous American population was killed by smallpox from their infected livestock.
Braveheart
Let's forget the fact that kilts weren't worn in Scotland until about 300 years after William Wallace's day and just do some simple math. According to the movie, Wallace's blue-eyed charm at the Battle of Falkirk was so overpowering, he seduced King Edward II's wife, Isabella of France, and the result of their affair was Edward III. But according to the history books, Isabella was three years old at the time of Falkirk, and Edward III was born seven years after Wallace died.
Elizabeth: The Golden Age
In 1585, when the movie takes place, Queen Elizabeth was 52 years old - Cate Blanchett was 36 when she shot the film - and was not being courted by suitors like Ivan the Terrible (who was dead by then). And though the movie has her rallying the troops at Tilbury astride a white steed in full armor with a sword, in fact she rode side saddle, carrying a baton. She was more of a regal majorette than Joan of Arc.
2001: A Space Odyssey
According to this film, in year 2001 we would have had manned voyages to Jupiter, a battle of wits with a sentient computer, and a quantum leap in human evolution. Instead we got the Mir Space Station falling from the sky, Windows XP, and Freddy Got Fingered. Apparently the lesson here is that sometimes it's better when the movies get the facts all wrong.