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.
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)
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)
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)
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)
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.
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.
Continuity mistake: As Thornhill backs away from murdered Townsend, the knife in his hand changes from point down to point up. (00:38:10)
Continuity mistake: When the aircraft crashes into the tanker, the position of the aircraft, relative to the edge of the road at impact and afterward, changes several times.
Answer: More than likely, they felt that Roger would be dead and they would not be found out. The fact that he survives their DUI plot and returns to the house with the police only serves to makes him look more suspicious and guilty. It's to move the plot along, nothing more.
ChiChi
The bigger plot hole is, if Van Dam really believes Roger is Kaplan, why would he think that Roger would bring the police and go through the trouble of preparing "Mrs. Kaplan" to make the police think he's crazy? If Roger really was a spy, he doesn't need help from the police and would have just disappeared instead of retracing his steps. So if Van Dam anticipated the actions taken by Roger, he must believe at some level that Roger is telling the truth and would have looked deeper into it.