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Continuity: When Carl is diving to retrieve the lost nuclear bomb, the Soviet submarine crashes into his air hose assembly and it gets caught in the bow dive planes of the submarine, which is accurate, since all Soviet submarines, including those of the era in which the movie is set, are designed with bow planes. Once the cables work themselves free, we see the submarine sailing away, its dive planes are mounted on the conning tower, which is where U.S. submarines during the 60s-90s had their dive planes mounted. No submarine is equipped with both bow and conning tower dive planes, so the conclusion is that the movie-makers used a model of a Soviet sub for the "tangling" and "dragging" shots, and a model of a U.S. sub for the "sailing away" shot.
Factual error: When Billy Sunday goes into Pappy's cabin, Pappy is shining some medals. As Pappy is giving his speech about things not mixing, one of the medals displayed is a Meritorious Service Medal. This medal was not introduced until 1969, long after this scene was supposed to have taken place.
Factual error: When the black scuba is out for his first (or second) weekend off school, he is at the port watching a ship passing. The prow of that vessel is "bulbous bow" type, (a kind of "appendix" in the prow that increases the speed). The "bulbous bow" concept for merchant marine vessels didn't start up until around 1964.
Continuity: When Carl Brashear in the courtroom is walking with the 130kg heavy diving suit, he is taking the last step against Leslie Sunday. When he has taken the step he moves his left foot one more time as easily as if it was a pair of Nike Air shoes he was wearing, but earlier he almost couldn't take his 9 step.