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.
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.
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.
Other mistake: The second bus in the scene before the crop duster has no license plate.
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.
Continuity mistake: Thornhill and his mother go to Kaplan's hotel room, room 796, to check the place out. There are two telephones in the room and both phones have old-fashioned non-coil cords connecting the phone and the handset. But when Thornhill rings the buzzer to summon the maid you can see that the phone suddenly has a more modern coil handset cord.
Revealing mistake: When Thornhill is going up the driveway of the house near Mount Rushmore, there are shadows, even though it is night-time! These shadows are far too strong to be moon shadows and there is no source of artificial light that could be making them.
Deliberate mistake: One of Eva Marie Saint's lines in the dining car seduction scene was re-dubbed. She originally said, "I never make love on an empty stomach," but it was changed in post-production to "I never discuss love on an empty stomach." It is said that the censors felt the original version was too risqué.
Factual error: The date shown on the newspaper after Cary Grant is accused of stabbing the diplomat at the U.N. is Nov. 25, 1958. The subsequent outdoor scenes clearly show leaves on the trees and people dressed for warmer weather.
Plot hole: The villains try to kill Cary Grant by getting him drunk and sending him off a cliff in a white Mercedes. He manages to get back on the road and they follow him in a Cadillac limousine. Cary passes a parked police car which gives chase. He brakes to avoid a bicyclist and the police car rear-ends the Mercedes. Immediately a blue 1941 Ford rear-ends the police car. Where did it come from? It would have to have been ahead of the Cadillac and very close to the police car to do so, but it doesn't appear until the moment of impact.
Audio problem: Before the famous attack by a crop spraying aircraft, Cary Grant is set down in the middle of nowhere from a Greyhound Scenicruiser. Another bus, a Flxible BR-29, also stops briefly. The soundtrack of the Scenicruiser's engine is re-used for the Flxible.
Continuity mistake: When the plane crashes into the truck, the bystanders walking towards the scene have shadows, which change angles as you see the next shot. They took them both in the morning and the afternoon.
Continuity mistake: When Thornhill and Vandamm are seated in the café at Mount Rushmore, Vandamm clasps his hands together and rests them on the table. In the next shot, he has his hand flat against his body. In the following shot, his hands are back to being clasped.
Continuity mistake: In the scene on the train in the dining car, the flower arrangement on the table changes. When the shot is over Cary Grant's shoulder the flowers are pink carnations with four large leaves, but when the shot is over Eve Marie Saint's shoulder it's a smaller display of pink daisies with one leaf.
Factual error: At the end the couple are supposed to be returning from Rapid City, SD, to New York by train. The last shot shows a locomotive of the Southern Pacific Railroad. The word "Pacific" on the two locomotives has been covered over (but not on the baggage and four passenger cars visible before fade-to-black). The consist is a mix of SP paint schemes, all of which identify the railroad regardless of names on the side. It would have been easy for them to obtain SP train equipment in LA, but it seems odd that they would have bothered to modify the railroad name on the locomotives, particularly since it changes the name to the name of a another real railroad that didn't serve the US north of Cincinnati and Washington D.C.: the Southern Railway.
Other mistake: When Cary Grant steps down from the bus from Chicago prior to the crop-duster scene, naturally the bus is heading away from his departure point. Later, when he hijacks the car to effect his getaway, he makes a U-turn so that, instead of returning to Chicago, he is heading in the opposite direction.
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.
Blibbetyblip