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.
Factual error: The address of the art auction is 1212 N. Michigan. Michigan Avenue only extends north to the 900 block.
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.
Visible crew/equipment: When Cary Grant and his film-mother enter the elevator at the Plaza this is filmed through a glass-door and you can see the reflection of a crewmember in a white shirt who is crouching below or next to the camera.
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: 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.
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 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: 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.
Factual error: The 20th Century Limited train, where Grant and Saint first meet, is supposed to be heading west in the afternoon, past the Great Lakes which should be on the right side of the train, but the lakes are visible through the windows on the left, so the views were shot heading east, and the supposed lowering late afternoon sun would in fact be the sun soon after sunrise.
Suggested correction: The fact that you can see the far shore of the "Great lake" tells me this isn't Ontario, Erie, or Michigan. I always assumed it must be the Finger Lakes, but I'm not familiar with the topography there. Just an assumption based on the knowing the Great Lakes and knowing that's not what was out the window.
Hitchcock was notorious for using location shots that looked good on the screen even if they were impossible in the context of the story. He depended on the viewers of the era not knowing the difference. Our ability to fact-check these things was never in his mindset. So I can completely believe the original poster's contention. (Cross reference - the Pacific Coast Highway shots in the out-of-control car scene at the beginning of the movie, even though this was set in Long Island.)
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: 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.
Revealing mistake: When Thornhill's cab arrives at the UN there's a couple behind him, a rear projection. Later on, when Valerian's car arrives at the UN the same projection is used, with the couple a mere two meters away from where they were before.
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.
Visible crew/equipment: Right before Thornhill gets shot by Eve, he's talking to Vandamm at the table. The camera moves backward. You can see its shadow on Thornhill's arm.
Visible crew/equipment: When Thornhill exits the phone booth 6 stage lights are reflected on the glass door.
