Revealing mistake: Alexander Dunning tells Agent Cooper to handcuff Frank. After Cooper handcuffs Frank it then cuts to a close-up of Frank's hand, showing Cooper giving him the key to the handcuffs. In this shot, we can see that the cuffs aren't actually locked in. He obviously is supposed to have locked the cuffs since he gave Frank a key to unlock them - if he faked it, why the key? This is most noticeable when Frank clenches and lowers his hand after being given the key, giving away the loose cuff. (01:41:25)
Character mistake: In the New Orleans scene in the cop car he calls in an officer down and describes the black vehicle as a Tahoe, but it's a GMC Yukon. He also describes the suspect as "Male, black suit and tie" but when he exits the vehicle they are both blue.
Character mistake: When Victoria is blasting away at the VPs limo, the security truck should block the limo from the machine gun. They just pull up behind it even though that vehicle is specifically in the motorcade to carry agents and to block threats even though it is usually not an armored vehicle.
Plot hole: When Moses comes to Sarah, a wet team arrives. Then, when Sarah gets free in the hotel and makes a 911 call, Cooper is on the plane and talks on the phone with someone about cheques and gets a notification about the 911 call. He comments why no one thought about that and asks the pilot to turn around. So who sent the wet team earlier?
Continuity mistake: In the beginning Frank wakes up at 6am. He walks downstairs and the clock in the kitchen shows 7am.
Suggested correction: I considered the 7:00 AM time on the clock as representing the passage of one hour since Red got up. Red is shown leaving his bedroom, walking down the hall while putting on his robe, and possibly entering the bathroom or other room. Perhaps more significantly, when Red enters the room where the later time (7:00) can be seen, he is no longer wearing the robe plus has a cup of coffee with him (which took time to prepare). After this, he goes to another room to exercise. These activities probably are his ways of passing time until he can call about his pension check - the office might not open until 8:00-9:00 AM.
Factual error: When Gabriel Singer gets shot in the control tower during the day it would've been impossible to see human beings in there. All towers have a severe and reflective coating for daytime and with the sun on them you cannot see into them. So there is no way the guy in the helicopter could've seen him to shoot him.
