Factual error: The Monet stolen by TC is credited as being the first painting in the Impressionist school, a clear reference to Monet's "Impression: Sunrise" of 1872. However, the painting shown during the entire film is "San Giorgio Maggiore by Twilight," a 1906 Monet painting.
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)
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)
Factual error: When Russo and Brosnan are talking at the meal when they first met it is said that he "went from Glasgow to Oxford on a boxing scholarship". Oxford have a firm policy throughout the colleges of only offering academic and music scholarships.
Factual error: The Trojan horse is transported into the vault area by a motorised pallet jack. The pallet jack operator is behind the load and his forward vision is totally obstructed by the load. The operator should have been in the front (as required by OSHA). Also they must turn the load around in the narrow hall because later the operator is in the front.
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.