Continuity mistake: The car's inspection certificate is dated 07 2006, at the end of the movie when they drive off it's 08 2006. (00:06:28 - 01:57:57)
Continuity mistake: During Claire's post mortem, Doug takes a picture of her face. Her head was tilted towards the camera as were her eyes. When the picture comes out, it shows that she was faced away from the camera. (00:18:20)
Continuity mistake: When Clair is on the post mortem table being examined, look at her eyes. When her head is turned to the right her eyes are as well but when they turn her head to the left her eyes are left as well. Another scene shows her eyes appearing to be looking straight up. Unless her eyes were manually moved they wouldn't change positions after death. (00:18:20)
Continuity mistake: The rig Carlin is holding repeatedly switches between his left and right hand. (01:06:25 - 01:07:25)
Continuity mistake: When the red Ford Bronco is driven off the ferry (ignoring all the impossible factors required for it to successfully do so), it flips through the air and contacts the water going backwards, its right/rear tail light entering the water first. The closer next shot shows the truck sink somewhat as it continues to flip backward, onto its roof. The very next shot shows the interior of the truck as it enters the water right-side up and going forward as water rushes in. Denzel lurches forward and down, driving his face into the water that is about waist level. The next exterior shot is underwater as the truck now lands in the water, nose first and right-side up. It then begins the expected slow dive, nose-first. (01:52:30)
Continuity mistake: When Denzel Washington is taking off his clothes to be sent back in time. We see him take off his blue shirt. The shot then changes to look at the other guy and we see him take off his shirt again.
Continuity mistake: When Carlin goes to the dock he gets out of the Hummer carrying a gun. The gun changes hands between shots as he is walking.
Continuity mistake: When Denzel makes a u-turn and a red BMW crashes, the front gets smashed, but a frame later, from a wide angle it's perfect.
Continuity mistake: In the scene where Doug and the terrorist are talking about 'destiny', the terrorist has a tiny bit of his cigarette left. But when the camera quickly shows a shot of Doug's face and then comes back to the terrorist, his hands are almost in exactly the same place, but the cigarette is gone.
Continuity mistake: In the beginning, the Pepsi can moves around between shots.
Continuity mistake: When the cop woman drags a timeline, the drawing keeps changing all the time. Watch the shape of the "X" for example.
Continuity mistake: When Denzel first leaves in the hummer he is on the Westbank heading toward the CCC Bridge to the Eastbank (downtown New Orleans). Once he makes it to the bridge, he's suddenly headed back to the Westbank (away from downtown) but on the Eastbank bound span (bridges are one-way), only to exit the freeway and suddenly appear on the Eastbank and cause the accident with the 18-wheeler.
Continuity mistake: While watching Claire in the shower she turns her head to her right. A frame later, from a new angle, she repeats the movement.
Continuity mistake: The colors of the lightbar on the ambulance stolen by Agent Carlin keep changing. Sometimes they are red and clear, other times they are red and blue (which wouldn't be allowed in Louisiana - blue is reserved for law enforcement).
Continuity mistake: When Denzel Washington is underneath the bridge, he uses his left hand to swipe the substance, while Val Kilmer holds his right hand. However, in the next shot, he sniffs his right hand to determine what the substance is, while his left hand appears to be in his pocket.
Continuity mistake: When investigating Claire's apartment Doug closes the bedroom door to reveal the mural. When listening to the answerphone the door is closed. Doug makes a note, next shot both doors are wide open.
Answer: Not a definitive answer, just an "allowable" excuse for this... The time surveillance team is testing their equipment and rigs and it's merely a coincidence that they happened to be in this area at this time.