Sydney wants Irving to flee with her to Estonia or Romania. At the time Estonia was solidly part of the Soviet Union and Romania was a brutal Stalinist dictatorship. Neither would have been necessarily impossible to get to but seem dangerously impractical destinations for Sydney to even consider, even with the law on her heels.
When Irving and Carmine are outside The Plaza Hotel you can clearly see the Bergdorf Goodman Men's Store across the street on the east side of Fifth Avenue. The Bergdorf Mens store did not move there until 1990.
At the tape-recorded interview in Amado's office, a suitably vintage cassette recorder is used, however the cassette tape is of a design used by TDK in the late-'90s/early-2000s, and the microphone used most closely resembles a model produced by Radio Shack in the '80s.
When Sydney and Irving discuss bank interest rates in his office shortly after first meeting in the early 1970s, Sydney curses Jimmy Carter for 12 to 18 percent rates, but Carter isn't President yet at this point in the story.
In the opening scene, Irving Rosenfeld sets his comb-over with L'Oreal Elnett hairspray. While Elnett did exist in 1978, it was banned in the USA at that time. Elnett became available in the USA in 2008, when a packaging change allowed for Target to have a highly publicized exclusive Elnett US launch.
When Sydney walks through the police station after being arrested in 1977, she passes FBI wanted posters for Patricia Hearst, who was apprehended in September of 1975, and Nancy Ling Perry, who killed by police in a shootout in February of 1974.
When Irving reflects on his childhood with his father's glass business (1950s), the windows being shattered are made of safety/tempered glass (which crumbles into granular chunks). This type of glass was not in standard use until 1966 and when we flash forward to the '70s, Irving picks up a piece of broken glass that is not safety glass.
When Irving and Sydney are discussing Duke Ellington, she states that he died this year, that would have to make it at least May 24th of 1974 the day he died, yet the party in Long Island was in January.
(at around 1h 12 mins) Irving's automobile is introduced as a blue 1977 Cadillac Eldorado. However, when Irving and his wife meet Mayor Polito and the casino financiers, Irving pulls up in a 1978 Cadillac Eldorado of the same color. You can tell it is a 1978 model because it has a thicker, boxier grille.
As the meeting between Victor Tellegio and the Sheik concludes, Tellegio tells Carmen that if they're serious, they'll transfer ten million dollars into an account and gestures with his hands. The camera angle changes and Tellegio repeats the same lines of dialogue with the same hand gestures but Tellegio's speech continues beyond what he said in the first shot, indicating it had been shot from two different cameras but not edited correctly.
According to Christian Bale, much of the movie was improvised. So, during the shooting of the film, he noted to Writer and Director David O. Russell, "You realize that this is going to change the plot greatly down track." To which Russell replied, "Christian, I hate plots. I am all about characters, that's it." This 'technique' of Russell's caused the characters to hesitate at times when speaking and often created extended pauses or silence in the dialogue which gave the plot of the film a very amateurish 'feel' because of the improvisation.
The goof items below may give away important plot points.
When Irving collapses after leaving Carmine's house, after telling the latter that the FBI are involved in the scheme they are working on, Sydney runs towards him, coming out of the blue car. As she comes out of the car she leaves the door of the car open at such an angle that it could not possibly close by itself, nor does she close it, nor does she push it in any way to make it close. Yet, we hear the door closing, and when she takes Irving back to the car it is indeed closed.