Continuity mistake: When Tommy Lee Jones pops into his local Irish boozer for a pint of what is supposed to be Guinness, it is clearly not a pint, it is clearly not Guinness (flat Coke, more like), and the level of said beverage in the glass fluctuates wildly and out of synch with TLJ's swigging.
Plot hole: The premise of the entire movie's main character, played by Jeff Bridges, is flawed, as it would be impossible for anyone to pass a major metropolitan police department's pre-employment screening background investigation for the position of police officer if that applicant uses a fake name and identity. Part of the investigation includes sending off the applicant's finger prints to the National Criminal Information Center, a polygraph test, a records check with vital statistics to verify place of birth, Dept. of Labor to verify past employment, Social Security Dept. and also interviewing (sometimes in person) your friends you've listed and your current and former neighbors of where you now live and use to live.
Deliberate mistake: When Jimmy Dove responds to the bomb in the computer lab at MIT, the camera shows the boyfriend on the floor, dead from a self-inflicted gunshot wound to the head. The pistol is lying next to his head and it is a 1911-type .45 automatic with the hammer down, not cocked. When an automatic pistol is fired the slide automatically cocks the hammer as it moves back. This will happen even if there is only a single bullet in the gun. The hammer should be back in a cocked position.
Continuity mistake: After Max is killed, Jimmy Dove is in the hot tub. As he speaks with his wife he fires one shot into the water, then tosses the gun onto the deck. The Glock is shown with the slide forward and the barrel pointed to the viewer's right. When his wife leaves moments later, without anyone touching the pistol, it is shown with the slide locked to the rear and the barrel pointed to the viewer's left.
Revealing mistake: As Kate drives the Jeep downhill near the end of the movie and swerves the Jeep quickly to avoid another vehicle, her head and body remain upright throughout the maneuver. Of course she was no doubt in front of a blue screen, or on a set, so there was no violent maneuvering to react to.
Continuity mistake: Near the end of the movie when Kate is in the Jeep careening down the hill, she swerves to avoid rear-ending a late 1980s Oldsmobile Cutlass Supreme, yet in the next shot, the Jeep is moving past a mid 1980s Ford Thunderbird. In subsequent shots, you can see the Thunderbird still behind the Jeep further up the road.