(at around 16 mins) When the C-130 overflew the rescue chopper crash site, they chose to use the sound from a multiengine reciprocating aircraft such as a DC-3. The proper sound would have been from turbine powerplants; the C-130 has turboprop powerplants and they do not sound like reciprocating powerplants.
(at around 53 mins) When Jake is in the bar, at the table with Emily, she gets up to leave, holding her purse and jacket, Jake then gets up to ask her to dance, and there is no sight of the jacket or purse.
(at around 45 mins) As Randall is stepping out of the "hypothermia pool" to speak with the other training officers, the camera briefly cuts back to Fischer inside the pool. Look quickly and you will see Randall in the foreground - even though he is supposed to be climbing out of the pool, and is standing outside of the pool when the camera cuts back again.
During Jake's first official rescue of the canoeists in the cave, a lifeguard's flotation briefly appears in the opening of a small side cave behind Randall when he preparing to bring out one of the canoeists.
The entire area is blanketed in snow when Fisch and Randall arrive at Kodiak, but in the cave rescue scene there is no snow anywhere on the cliffs or on the bluff above the cave. That is because the scene was filmed near Cape Disappointment, Washington, not Alaska.
(at around 1h 12 mins) The "Squid Bar" Hodges and Fisch go to is filled with Sailors in the utility uniforms, along with various officers and chiefs. Sailors are not allowed to drink in the utility uniform or even go into a bar.
Several times, we're told that Fischer was offered swimming scholarships by "all the Ivy League Schools." Ivy League schools do not offer athletic scholarships. Also, the Ivies haven't had first rate swimming programs since Yale in the early sixties. An outstanding high school swimmer would be receiving scholarship offers from schools in conferences such as the SEC or Pac 10.
The motto quoted in the film and mentioned in the closing title card - "So Others May Live" - is not an official motto of the Coast Guard. However, Many members of the Coast Guard say this is an unofficial motto that many use anyway. It is very close to that of the creed of the Pararescue Jumpers, "That Others May Live", part of US Air Force Special Operations Command.
The actual motto of the Coast Guard is "Semper Paratus", Latin for "Always Ready" or "Always Prepared".
In the training room, after Randall is discussed as the record holder, the students are brought to attention. All but one is standing to attention, he is in the back on the right hand side, standing at ease. The person standing at ease is actually an instructor and would not be called to attention with the class.
(at around 52 mins) When Jake meets Emily to give her half of his bet winnings, they are at the bar. There are even other patrons leaning up against the bar right next to Emily. When she holds up Jake's napkin and says, "This is just a picture of your pants," they are sitting at a table and then get up and walk over to the dance floor. There is some time passing between the two shots.