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)
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)
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)
Revealing mistake: During the glider scenes, they use aerial shots of stuntmen in a real glider and then a prop/dummy one with a blue screen for the close up shots of Thomas and Catherine inside the cockpit. The actual glider has a second rear cockpit control box that you can see in all the aerial shots.The rear pilot is at least 5 feet back from the front one. In the close-ups, with the prop/dummy one, the rear cockpit box is gone and they show Thomas actually leaning on Catherine's back. The actual glider also has a much larger wingspan, fuselage, and cockpit. Again, you see this during the take off shots and all the aerials. When they land in the field and crawl out of the dummy/prop, the tail is only about 5 feet back from the cockpit. (01:04:50 - 01:06:05)
Revealing mistake: During the glider scene when they show the close up of Catherine's face with Thomas leaning over her from behind, waving his hands, if you look at her sunglasses you can see all the white screens, the camera crew, the reflection of his hands, etc. You can see this every time they show a close-up of them from the front of the glider. (01:06:00)
Revealing mistake: During all shots of the actual glider, you can tell very obviously it is two stunt men in very bad, fake looking wigs. The one up front, "Catherine" now makes her look like a football player. He also has on a long red wig that goes to his shoulders, which is longer than Catherine's hair. In one aerial shot, the stuntman sitting in the rear, which is supposed to be "Thomas", has this dark black bowl cut wig that is parted down the middle and slicked to his head. (01:06:05)
Revealing mistake: In the opening scene, the outdoor setting is supposedly an unusually hot autumn day. However, outdoor shots show several people with jackets, long coats, and almost no short-sleeve tops or shorts.
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.