In the beatnik club, Duke Ellington's "In a Sentimental Mood"
plays in the background. The scene takes place New Year's Eve 1958, but uses the Impulse! recording from Duke Ellington and John Coltrane, which wasn't released until 1962.
Dr. Bronfenbrenner's film of Norville's psychiatric session begins with a SMPTE Universal leader that counts down from 8 to 2. This type of film leader wasn't developed until the 1960s. The film would have actually had either an Academy leader counting from 11 to 3 with black film in between or one of the brand new (for the late 1950s) SMPTE Society leaders that counted from 11 to 3 but had focusing cross hairs, rather than black film, between the numbers.
Near the end of the film, when Norville walks out of the juice bar, one of the album covers hanging on the wall on the right hand side of the screen is Tom Lehrer Revisited (Lehrer Records TL 201). This is supposed to take place on December 31, 1958, however that album was not recorded until November 23 & 24, 1959 (side 1) and March 21 & May 4, 1960 (side 2), and its cover photo was not taken until June 29, 1960. The album was released in the second half of 1960.
When Norville is in the mail-room, telling the old mail-sorter about his invention, Norville holds up the drawing, and you can see Norville's lips are not moving when he says his line: "You know, for kids."
When the boy is playing with the hula hoop, a crowd of kids runs up to him. You can see a boy with a green shirt and yellow suspenders to the right of screen. But when the crowd runs off, this boy is to the left of the screen.
When Norville is carrying Amy into his office, he opens the door with is buttocks, where the sign man is painting his name on the door. When he bends over to lay Amy down, there is no paint on his buttocks. But two scenes later he bends over again, the writing from the door can be seen on his pants.
The time period of this film is the last month of 1958, i.e., December. Yet, when the Hula Hoop hits the stores, the Toy store shown shows nothing indicating Christmas is only a few weeks away, and all but a few of the boys shown in that sequence are wearing short-sleeved shirts, impossible in a New York December.
The goof items below may give away important plot points.
In the first few minutes of the movie, we see a wide shot of the Hudsucker building from blocks away. However, when Waring Hudsucker jumps, there is clearly another building directly across the street. If that building were there, we could not get such a clear view in the first shot. Not only that, but as Hudsucker is falling, the scene cuts to a secretary sitting by the window, and again, the building across the street is gone.
When Waring Hudsucker jumps from the conference room they say he "drops 44 floors...", "45.", "Counting the mezzanine.", we see from all the outside shots that this is the first floor under the clock. When Norville Barnes enters the elevator to take the blue card to Sidney J. Mussburger's office he asks to go to the 44th floor but since Sidney J. Mussburger's office has the "3" from the clock visible it must be 2 floors above the conference room floor, per the above mentioned outside shots, which would be the 46th floor assuming the conference room is on the 44th. And then at the end of the movie Moses says "Norville Barnes climbed waaay up to the forty-fourth floor of the Hudsucker Buildling... there was a man who jumped from the 45th floor", his office was on the 46th floor and assuming he meant the sign painter, he actually "jumped" from the upper level of the two story high 46th floor and would have landed on the conference room ceiling.