Revealing mistake: In the shooting scene in the Mount Rushmore cafe, a boy in the background puts his fingers in his ears, because he knows the gun is about to be shot.
Continuity mistake: When in James Mason's house in South Dakota, Cary Grant writes a message to Eva Marie Saint on a matchbook from which several matches are missing. When Ms. Saint subsequently reads the message the matchbook is full.
Continuity mistake: In the scene where Eve writes on the note pad she doesn't write enough. Later you can see much more to the address. (01:20:50 - 01:25:15)
Continuity mistake: When Eva Marie Saint grabs the statue from James Mason and bolts, the spies chase after her. They had no way of knowing she would do such a thing, and yet they are suddenly equipped with long, heavy-duty flashlights.
Factual error: When Cary Grant is attacked by the crop duster, he is supposed to be between Chicago and Indianapolis. Unfortunately, northwest Indiana looks nothing like this. The scene in the movie has miles and miles of treeless landscape. Even in the flattest farmlands of Indiana, there are many visible groves of trees.
Continuity mistake: In the scene where Cary Grant is hanging off Mount Rushmore, and Eva Marie Saint falls and grabs Cary's back pocket, she rips it but in the next scene the pants are un-ripped.
Continuity mistake: In Eva Marie Saint's hotel room, after the crop-dusting, Cary Grant attempts to copy a note she has removed from a pad, by scratching with a pencil point - horizontally. In the close-up of the note, it has been scratched vertically.
Continuity mistake: The first scene that they show of the train going up the Hudson River is wrong. The hills in the background are much further up the river past Croton NY. When they are eating the train is back down near the wider part of the Hudson around Tarrytown NY. (00:45:25 - 00:47:00)
Visible crew/equipment: Cary Grant is crouching in his bunk bed in the train. The lights of the studio get reflected on his broken sunglasses when he shows them to Eve.
Continuity mistake: Near the beginning of the film, as Thornhill has arrived at The Plaza and is opening the door of the taxi to exit it; visible through the rear window is a red and white Ford taxi, parked behind Thornhill's taxi. Instantaneously, in the next shot taken from the sidewalk, he is shown completing his exit from the cab. Now however, there is a light colored Dodge taxi cab parked directly behind them. (00:04:20)
Continuity mistake: While Thornhill and his mother are investigating Kaplan's hotel room, he decides to try on one of the man's suits from the closet to see if he and Kaplan are similar in stature. He tries on a navy or dark colored suit coat and then holds the pants up against himself to check the length. In the close up of the trousers, they are now a light gray in color. The camera returns to a full shot and, once again, he has the darker trousers in his hands. You can also see the light gray suit in the closet behind him. (00:31:25)
Factual error: In New York, a newspaper headline reads, MANHUNT ON FOR UN KILLER, with a smaller headline below reading "Nixon promises West will remain in Berlin". The next night Eva Marie Saint reads the crop-duster plane story in the Chicago Sun-Times. Since the Sun-Times was a morning paper and the accident occurred in the afternoon, she must have been reading the following morning's edition. But a smaller headline underneath carries the same story: "Nixon promises West will remain in Berlin"; two days later, it's old news.
Continuity mistake: While Roger and Eve converse before dinner on the train, Roger's glass mysteriously changes places several times. When shot from behind, he's holding the glass; from the front, it's on the table.
Continuity mistake: Cary Grant, running through Grand Central Station after fleeing from the UN Building, has brown shoes, then black, then brown.
Continuity mistake: When Cary Grant writes a message on a matchbook and Eve Marie Saint reads it, the number of lines in the message changes. He writes 3 lines, she reads 4 lines. (01:58:30)
Other mistake: The second bus in the scene before the crop duster has no license plate.
Continuity mistake: In the very first scene when the train pulls out of Grand Central Station you can see that the sun is setting. Then when they sit down to eat it's the middle of the day again. (00:45:25 - 00:46:50)
Continuity mistake: In the last scene before Chicago the train is going through the tunnel at Breakneck Pass north of Cold Spring NY. When they're going back home at the end of the movie theoretically they should be going the same way but they're not. The final tunnel shot in North by Northwest is the Southern Pacific railroad tunnel between Santa Susana and Chatsworth California. (01:00:10 - 02:16:05)
Plot hole: When Eve is being led out to the airplane, she bolts when two shots are fired from inside the house (presumably Anna firing at Thornhill, not realizing that the gun was loaded with blanks). Moments later, she jumps into a car driven by Thornhill, and they escape together. There is no way Thornhill could have gotten there that fast, given that he was in the house only seconds earlier.
Continuity mistake: On the patio outside the restaurant at Mt. Rushmore the professor goes inside first. His shadow falls upward to the right, there is no glare on the upper windows and no building shadow on the ground. Thornhill walks in seconds later and his shadow falls downwards to the left, there is a strong glare on the windows and the building casting a shadow on the ground.
Answer: Not really. You could (and at some hotels are still able to) keep your room number private or you could not - i.e. you could ask the hotel staff to keep your number secret from strangers, or you could ask them to tell anyone who might ask. Not having seen this movie, I don't know how likely it would be in the situations you speak of that the hotel guest would choose the latter option- it might be a mistake.
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