Factual error: One of the scenes shows the buildup of American readiness as tensions mount. A B-52 bomber appears carrying four white menacing-looking missiles under her wings. Such missiles can easily be identified as the Hound Dog type, which was under test in the mid-60s but never became operational. America would never deploy those missiles either for real or for training, because the project was scrapped before even being accepted into the Air Force.
Factual error: The U-2 was built to fly on the edges of its flight envelope: too slow yet too high. The pilot had a very difficult time keeping it level and straight so it wouldn't stall. In the film, a U-2 evades missiles by jinking up and down as if it were a fighter, until a final missile strikes it down. That aircraft would never have been able to even climb away from the first missile, all it could do was sit and wait for hell to reach up. Those aerobatics were completely out of the question for a U-2.
Continuity mistake: The crew on the destroyer imposing the blockade initially saw the two Russian freighters on their radar screen, and then they saw a third 'blip' on the radar screen as the ships approached the blockade. The radarman identified the first two blips as surface ships, and the third blip as a submarine. Unless that was a sonar display, they would never have seen the submarine. The third contact would have been made by the ship's sonarman, and the two surface contacts would have been identified by the radarman.
Continuity mistake: The Pierce fires its guns to clear the barrels of the already loaded starshell. This is technically correct but when the starshells burst there are four of them i.e. the same as the number of gun barrels on the Pierce. Earlier however, when the Pierce first confronts the Grozny, there is a shot in the 5in gun turret showing at least one barrel being loaded with practice shell (i.e. solid shot). The Captain reports to the Admiral that the guns are loaded. (01:30:15)
Continuity mistake: When the U2 gets shot down, the missiles launched appear to be truck launched SA3s. The only missiles that the Soviets possessed at the time that could get anywhere near the altitude of a U2 are SA2s, which are taller than telephone poles.
Continuity mistake: In the final interception of a Russian ship, the vessel is described as a tanker, but in reality, the ship shown on the screen is a freighter.
Factual error: On the scene where the 2 low-flying RF-8 Crusaders are being shot at, the Cubans and Russians are aiming what supposed to be a mobile SAM launcher, that resembles SA-3. As far as I know, at the time the Soviets had no mobile SAM, until the introduction of the SA-6 in 1966.
Visible crew/equipment: At 54:20 they show the commander of the ship looking through binoculars they then show a shot of the two ships taken from what was supposed to be the American military ship. On the far left of the screen is the American ship visible is somebody in a blue life jacket just like the military personnel however there is someone else who is sitting down with long hair and a beard (most likely a crew member), that is obviously not in the military. (00:54:15)
Visible crew/equipment: In the beginning of the film when the family is gathered around the table for a meal, the crew, cameras, and lights are visible in the reflections of the window.
Continuity mistake: In a scene where O'Donnell meets with Kennedy in his private quarters, Kennedy unties his tie and leaves it hanging around his shoulders. A moment later it has somehow crawled partially back up under his collar.
Factual error: In the last scene in the film, Costner and his family are seen having breakfast before going to mass. The problem is, they are portrayed as a strict Roman Catholic family. No strict Roman Catholic family would eat breakfast before going to mass at that time. In 1963 you were not even allowed a glass of water from the moment you woke up till you took communion at mass.
Continuity mistake: There were several flash frames in the film - the most obvious was when an eight appeared in an atomic bomb shot. The eight was from film leader and the result of a bad edit.
Continuity mistake: When the President and advisors are sitting around a large table and communicating with the captain of the ship at the beginning of the blockade, when viewed from the front, the President clearly has his right hand up to his chin while concerned and deep in thought. As the camera angle switches to his back several times, his left hand is on his chin. Front angle right, rear angle left.