The pay grade of Senior Chief Petty Officer was not created until June 1, 1958. Service-wide exams were conducted for eligible Chief Petty Officers on August 5, 1958. The first batch of Chiefs to be promoted to Senior Chief occurred on November 16, 1958.
In the initial scene in the barracks a modern fire extinguisher is seen hanging on the wall in the background. At this time the barracks would have had either a simple water bucket or a more primitive extinguisher.
When Carl Brashear is in his hospital bed, ostensibly in 1968, both Admiral French and Captain Hanks are shown wearing the Navy Sea Service Deployment Ribbon. This ribbon was not established until May 1980, and made retroactive only to August 1974.
At the beginning of the final diving exercise, the big navy clock says it's 2 in the afternoon. A few hours later we see a diver coming out of the water Sunday says that he's been down for over four hours. But the subsequent scene shows the nervous Jo waiting for a phone call from Brashear, and the clock says that it's 3:40.
When we first meet Jo, she says that she hasn't seen her father since she was 9 years old. When she comes to the navy bar, she tells Carl Brashear that she pulled her father out of enough bars and she couldn't do it anymore, etc. It's highly unlikely that she was pulling her father out of bars at such a young age.
While Carl is putting his wedding band with his tags, he takes off his ring, removes his tags from around his neck and puts them on his chest and you can see there is nothing around his neck, in the next shot you can clearly see the tag chain around his neck, next shot is him putting the ring on the tags and putting the chain back around his neck.
Near the beginning of the movie, young Carl's father is plowing a field. While insisting that young Carl go to school as opposed to helping him plow a field, Carl's father places the long leather rein that controls the mules pulling the plow over his head with half the rein going under his left arm. In the close up shot that immediately follows Carl's father having placed the rein under his left arm, the rein appears around his neck with no portion under his left arm.
Everybody in the movie had embroidered name tapes on their uniform shirts and trousers. This is a relatively new uniform item and wasn't fully implemented until 1999. During the period the movie represents, names would have been written in black india ink using a stencil. (Sometimes just scrawled in really bad penmanship with a marker)
The anchor device denoting rank on a chief petty officer was shown being worn parallel to the front edge on a long sleeved shirt worn with a tie. It was actually worn pointing towards the tip of the collar only on a short sleeved shirt with no tie.
Carl Brashear recovers a Hydrogen bomb but the newspaper, that Billy Sunday reads, says "A-Bomb recovered". Hydrogen bomb is called H-bomb whereas an Atom Bomb is known as A-bomb, both in civilian and military parlance.
The news report in the film states that three nuclear bombs were lost in the accident, with two being immediately recovered while the third was being searched for. It also stated that they were 50 Megaton warheads. In reality, there were four that were lost in the accident, with three being located almost immediately and the fourth being the one searched for. Also, the bombs were Mk28 bombs, of which the largest possible yield was 1.45 Megatons, not 50 Megatons, a bomb size in which no country has ever yielded, save for a 50 Megaton prototype bomb, called the Tsar Bomba, which was tested by the Soviet Union on Oct 30, 1961. Also, they were gravity bombs, which fall freely to earth, not warheads, which are carried on missiles.
Jo's character seems to go wherever the script wants her to - it doesn't seem likely that a young black girl would walk alone into a rough Navy bar in a racist atmosphere, and it's hard to imagine how she showed up at Carl's dive test unescorted.
In the scene where Sunday is found guilty of assaulting Hanks and is handed his sentence, the camera zooms into Sunday's face and Sunday blinks backward (in reverse - twice). The scene was possibly shot with a view of Sunday's eyes and the camera was supposed to zoom out to see him sitting in the chair, and edited to play in reverse for the final cut.