Watch carefully the scene where they are celebrating New Year's Eve. There is a banner up on the wall which says "Happy New Year 1956". The movie was set to start in 1962, I believe, thus putting this banner in the "BIG" slip-up category. It should have read 1963.
This isn't a slip up. If you watch the New Year's party scene on the DVD with the director's commentary (like I just did a few mins ago), you'll hear them talking about this scene and how they wanted to do a transition from the 50s to the 60s. At the end of the movie, they show that in 1968, he was the first African-American to become master chief. I don't think that this man accomplished all that in just 4-5 years (if your slip up is correct). Also, if it were true that the movie started in 1962, at the New Year's party, his wife tells him that she's pregnant, right? That would mean that at the end of the movie, when he's taking those breath-taking 12 steps in the heavy divers suit, his son would be about 5 years old. If you look at the boy, he looks around 8 or 9. Sorry