Continuity mistake: The number of plastic self-adhesive stickers attached to the left side of the contract change. After the contract is flopped onto the table and just before Crown signs it, the first side view (very quickly) shows 3 stickers attached to the left side of the paper with red on the bottom. Crown pulls the pen from his jacket. No one touches the contract. The second view shows 2 tabs attached at the same place with red on the top. (00:00:08 - 00:07:58)
Revealing mistake: At the beginning, when Thomas Crown is in his car, getting ready to get out and walk up to the museum, they show a close up of his wrist watch to set the time; it says 9:08 a.m. A few minutes later when he is sitting inside the gallery eating his breakfast croissant, if you look carefully at his watch, it shows 4:30. In various other shots, it also shows 5:25 and 6:30, indicating at least 2 hours spent shooting the gallery scene. (00:02:25 - 00:04:00)
Continuity mistake: When he first sits in the museum, his watch shows about 4:30. Next scene, as the truck pulls into the museum, the clock on the right wall shows either 1:30 or 6:05. Next scene as he walks into the office and says good morning, it's shortly before 12 noon. (00:02:29 - 00:04:00)
Continuity mistake: In the opening sequence, when Crown decides to walk and has "Jimmy" meet him at the office, he is almost run down by a truck. First of all, the truck has to skid to a stop, yet there is a car no more than ten feet in front of it when they change camera angle. Why was the truck going so fast in the first place. Secondly, when Crown motions for the driver to pass, the same car is still stopped directly in front of the truck, yet the truck speeds off in a way that it should smash into the car in front of it. (00:02:45)
Visible crew/equipment: When Crown nearly walks into the lorry at the beginning you can see cables of sort in the left hand mirror from the shot inside the lorry. (00:02:50)
Continuity mistake: In the first scene in the gallery when TC is admiring the Haystacks painting, he is preparing to eat a croissant. In the shot looking at him face on, he maneuvers the croissant into position in the bag it is in, so that he can begin to eat it. However, in the next shot - a longer distance profile shot - just after he answers Bobby's question, he is already chewing on a mouthful and starting to tear another piece off. (00:04:00)
Other mistake: Near the start of the movie when Thomas Crown enters his office for the first time, he uses a magnetic swipe card to access the elevator to his office. After he swipes the card, you can hear a 'ding' and the sound of elevator doors opening but you'll notice as he first inserts the card into the reader that the doors are already open. (00:05:30)
Revealing mistake: When Crown enters his office for the first time, he goes up to the window and looks at the river. Just as the camera points out the window, a bright light is seen to shine on Crown's face so that his reflection in the window is clear. (00:06:25)
Factual error: When the thieves are coming out of the Trojan horse, during a shot a pile of sawdust in a cone formation on the ground. This formation couldn't have formed since the thief was sawing a line in the horse, not drilling one hole. (00:09:05)
Continuity mistake: When the "thieves" are preparing the impressionist wing and inform Thomas Crown that the exhibit is closed, Crown responds with "It's only a quarter to five". Shortly after when the surveillance footage from the time of the robbery is being reviewed, the time stamp on the monitor says 5:54. I doubt that the whole robbery took 1 hour 9 mins, when the time in the movie was around 3 minutes. (00:17:40 - 00:28:30)
Character mistake: Thomas rips off his one glove as he makes his escape but then grips the bottom edge of the security door to slide under it. Smooth metal door would leave perfect finger and palm prints. I doubt detectives would miss prints on a door that was normally recessed away from the public. His prints may not be on file but would be easy to obtain once he was under suspicion. (00:19:20)
Continuity mistake: When TC steals the Monet, he places the painting in the briefcase and closes the case breaking the frame in half and thus ripping the edges of the painting. When at home he opens the briefcase and the frame is obviously crooked but when he holds the painting up to place it above the fireplace it is perfectly intact and the painting is not damaged. (00:20:10 - 00:22:30)
Continuity mistake: When TC opens the briefcase to steal the Monet, the thumb of his latex glove is ripped wide open, which would leave fingerprints. After he places the painting in the briefcase, the next shot shows him closing the brief case and the glove has magically repaired itself. (00:20:10)
Visible crew/equipment: When Thomas Crown exits the museum after having stolen the Monet, we see a couple of strange characters, probably crew members, standing on the ledge of the building just behind the cab he gets into. The man with the mask is the most interesting. (00:20:15)
Continuity mistake: When Crown returns home straight after stealing the Monet, the high external shot shows him open the door with his left hand. In the next internal shot he is opening it with his right hand. (00:21:25)
Plot hole: When Renee Russo was reviewing the tape showing the room with the painting to see who stole the painting, and the tape was blank because the heat generated from the suitcase hidden under the bench, why didn't she just back up the tape completely to see who put the suitcase there in the first place? (00:28:30)
Factual error: The man in the security room, when watching the thermal video footage states that the thermal cameras needs a difference of ten degrees (or something like that), thermal cameras operate within a range of two degrees so it would have to be a fairly rubbish camera and that's unlikely, seeing how it's guarding a Monet and all. (00:28:45)
Continuity mistake: Thomas Crown comes into the police station to identify the thieves. As he is getting ready to leave, Catherine eavesdrops on his conversation as she fixes herself a cup of coffee. She then watches him out the window, drinking her coffee, as he runs to his car. The paper coffee cup changes from a purple swirl design with a little bit of green to a blue and white, geometric border with what appears to be an Asian design. (00:32:30)
Continuity mistake: During the scene when Thomas Crown is sailing, you can see the weather change from very dark and stormy, to just cloudy, to blue sky and sun, depending on the shot. (00:36:30)
Revealing mistake: During the catamaran sailing scene, the boat jibes and Brosnan leaps across the trampoline and starts grinding on an empty winch, with no line in it. (00:37:40)
Answer: I believe that the Monet that Crown hides in his study is not the one that was stolen, it is a copy that he already had prepared. He can enjoy the copy knowing that the original (with the broken spreader bars) is also in his possession. The stolen original then goes to the forger who repairs the broken spreader bars, and then paints another painting (using water soluble paint) over the Monet, so he can "return" it to the museum 3 days later. It gets more complicated when he discovers that Russo is on to him so he has a second forgery made (even the edges forged to match) over the top of "Dogs Playing Poker." He doesn't know if it will be necessary, but given his research into his new adversary, he concocts this contingency. It is likely that he has many contingencies in place, but the "Monet with a ghost underneath" is the only one we get to see. Of course for my theory to hold water, there must be (or have been) that earlier forgery - unless it has been destroyed.
It's not the forgery that he takes out of the briefcase. Even if it were, he still put the Monet in the briefcase at the museum and would have had to break the frame to close the briefcase, thus also breaking the paint and tearing the canvas. The real answer is that it is just something that couldn't really happen, and the movie people don't want the viewer to notice.