Continuity mistake: His suit changes colour as he is running from the aeroplane. First blue, then brown, then back to blue.
Continuity mistake: When Cary Grant and Eva Marie Saint are riding in the dinner car on the train, there are several instances where the landscape/view as seen from their window repeats itself. Keep an eye out for a bridge they never get any closer to, and a section of tall iron work near the tracks they keep passing over and over.
Continuity mistake: When Cary Grant arrives at the hotel in Chicago, he sees Eva Marie Saint enter an elevator. By looking at the indicator light he sees that she has gotten off on the fourth floor. He would have had to have binoculars to see the elevator indicator light from where he was standing.
Continuity mistake: The top and back of Mount Rushmore looks nothing like they portray it in the final scenes of the film. This is a mountain, and they would need climbing equipment to get to the top of the mountain. Also there are no trees at the top, and the is a vault at the top of the mountain, behind the monuments. This vault was originally planned to be used as a museum, but Congress withdrew funding because of World War II, and the museum was never completed.
Continuity mistake: In the scene where Cary Grant runs out of the hotel and jumps into a waiting cab, the two bad guys giving chase immediately jump in the next cab and should be the very next vehicle right behind Cary Grant's. But when Cary Grant looks behind him through the cab window to see if he's being followed, the bad guy's cab has disappeared. We get a clear line of sight all the way back to the hotel and the car is not there. Instead we see a third cab pulling into the space they just left.
Continuity mistake: Cary's position was close to Eva and he almost kissed her, but then his position is farther away when he asks, "Shall I?" Also, in the kissing scene, when Cary is shown he is kissing her near an open door, but then they are against the wall again. (00:53:40)
Continuity mistake: When Cary Grant's character, Roger Thornhill arrives at the UN building in search of George Kaplan we see his shadow fall onto the door of the cab, as if the sun is behind him. But when you look at the pedestrians on the sidewalk, their shadow falls in the opposite direction. Simple mistake to make when using 3 point lighting in a studio environment to produce a rear projection process shot.
Continuity mistake: At Prairie Stop 21, after the second man arrives, shots on Thornhill show that his left arm is bent (the sleeve is folded), but the wide shot on both of them shows his arms at his sides. (01:09:45)
Continuity mistake: When Eva Marie Saint is going out to the aeroplane near the end of the film, it is dark as they walk away from the house, but by the time they reach the plane it is suddenly daylight.
Continuity mistake: When Thornhill is talking to the maid in Kaplan's hotel room, right after she identifies him as Kaplan, his arm changes from bent over his stomach to straight at his side. (00:29:45)
Continuity mistake: While drinking in the train Thornhill's hand is grabbing the glass or not, depending on the shot. The position of the glass to his face also changes.
Continuity mistake: When Thornhill is about to be run over by the truck his jacket is covered in dust. When he falls on the ground it's spotless, only to appear dirty a shot later.
Continuity mistake: When the truck is about to run Thornhill over, skidmarks running along the road appear/disappear randomly between shots.
Continuity mistake: When the plane crashes against the truck its wipers face the left side. A shot later they are facing the right side.
Continuity mistake: When the opening credits roll-in stating that the events are fictitious, there is a shot of a woman in blue exiting a building. When the credits end after Hitchcock's cameo, the same shot is re-used though this time from a couple of seconds earlier.
Continuity mistake: When Thornhill is served a Martini in the train, check the toothpick and you'll see it keeps moving around between shots.
Continuity mistake: When Thornhill makes a call at the train station the cord of the phone is curled in the wide angle but straight in the close-up.
Continuity mistake: When the CIA meets, the paper next to the professor changes positions between shots and a matchbox appears and disappears randomly.
Continuity mistake: When Thornhill is at the Plaza with his friends, a man grabs a drink and takes a sip while he places his left hand on the table. A shot later the glass is on the table and his left hand not.
Continuity mistake: Amount of people walking on the street changes drastically between the moment Thornhill is in the cab with his secretary to when he gets out.
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.