Plot hole: When Nick confronts the P.I. and yells "Why are you following me?", he looks inside the car and see the file with his picture on the front seat. The P.I. notices this and turns the file over. After Nick grabs his gun, the P.I. jumps out on the other side, raises his hands in the air and tells him somone hired him and runs off. As the P.I. runs off, Nick yells "Who hired you?" as he stands alone next to the car. He then waits a minute and walks off. If he really wanted to know more information, he could have taken the file that was right in front of him on the carseat. It makes no sense that he is desperate for information about the strange events that are going on, sees this file on him, and yet doesn't even pick it up or look at it.
Plot hole: At the end, when Nick is on the roof and everyone is behind the metal doors, you hear the saws trying cut through the door and the sparks flying. Immediately when they cut through, the doors swing open and everyone is standing right up front with Champagne glasses. No one is holding any saws or equipment or have any in front of them. Nor was there was any time for the people cutting the doors to run to the back to hide behind everyone else since the sparks fly right up until the doors swing wide open.
Deliberate mistake: Every time Nicholas enters his gate at home, it it daylight and then when he drives up to the house it is night. Either he has a 300 mile property or there is something else going on. It would appear that it is deliberate as it happens many times in the film. Symbolically, Nick is leaving the world as he appears with wealth and privilege (light) and entering the real van Orton world of solitude and sadness (dark).
Continuity mistake: In the scene at the restaurant, when the waitress runs into Michael Douglas and they both get wine all over each other, watch her shirt carefully. In one shot there are many wine stains all over. The next shot, there's not much. Then it goes back to having lots of wine stains.
Other mistake: After taking the CRS test, while talking with Feingold, Nicholas corrects his use of "unsatisfied" with "dissatisfied". Feingold responds, "That's right, you're a left-brain word fetishist." But earlier, on the phone with his ex-wife, Nicholas describes her growing family as "nucular" (as opposed to "nuclear" - the "cl" sound in the middle is key). Someone obsessive about word use would never make this mistake, it is classically seen as a sign of low intelligence and/or backward education.
Audio problem: When Michael Douglas catches up with James Rebhorn at the zoo, just as Douglas grabs Rebhorn he pulls him in close and brandishes his pistol. You hear the sound of the gun being cocked (i.e. the hammer being pulled back), but when the camera actually shows the gun, the hammer is not in the cocked position.
Character mistake: In the scene where Michael Douglas pulls out the CRS contract to show his lawyer, the page is blank and a surprised Douglas says "They used invisible ink." He should have said "disappearing" ink, which is not the same thing at all. Invisible ink doesn't show up unless treated with some form of catalyst such as lemon juice or heat. Disappearing ink vanishes after a period of time, which is what happened on the contract.