Continuity mistake: During the scene when Russell Crowe and David Caruso come to Meg Ryan's house while they are packing the "Good Faith" money, Meg Ryan is holding a cigarette. As she gets excited, she starts waving her arms around and running her hands through her hair. You can see the cigarette is not lit. She then goes into a closet to have a discussion with Peter's sister. While in the closet, she takes a drag of off the now lit cigarette. At no time does she stop to light it.
Continuity mistake: During the escape scene, when the missionary is cornered at the edge of the waterfall, there is a camera angle looking up at him from below. Though this shot is shown before his pursuers fire at him, he is already holding his wounded shoulder. Also, the scene makes it appear that the bullet knocks him over the cliff, but when his pursuers get to the place where he was standing they find a great deal of blood on the rocks. It seems the original idea was to have him take a shot in the shoulder and then choose to dive in, but to save time (or create the idea that his captors might think they had shot him to death) the scene was re-written in the editing room.
Continuity mistake: Russell Crowe is talking to his boss on a boat as it sails down the Thames. The boat passes under a train bridge and Blackfriars bridge. Check out the continuity errors as the camera switches from Russell to his boss throughout that scene. The men are standing next to each other and yet there are moments when his boss is almost under the second bridge and poor old Russell is still waiting to go under the first.
Continuity mistake: During the rescue of the hostage, one lone bad guy attacks them at the chopper. The hostage shoots him many times with his pistol. One shot, the camera is looking over the hostage's shoulder, you see the pistol's slide lock back (meaning it is out of bullets). In the next shot, with the camera in front of the hostage, the pistol is in normal firing position. He didn't have the time nor the magazine to reload the weapon.
Visible crew/equipment: Near the end of the film after Russell and his buddies have stormed the camp and rescued Meg's husband, Russell sets a claymore mine on the escape path. As soon as the bad guys get close to the mine, Russell detonates it. If you look above the two bad guys as the mine is detonating (just before the camera cuts to Russell and back), you can see the wires they are harnessed to.
Other mistake: When they're running away from the embassy because of the bomb threat he uses his key chain remote to shut off the alarm to his car. Then when they get in there's an envelope, the proof of life, on the floor. The sister asks how they got in. That's a good question. The alarm was on. How did they get into the car?