Deliberate mistake: The computer Felicity Huffman pushes off the desk is a dummy model and is not connected to anything. When she first pushes the computer off, the monitor only has a power cable visible. These computers were connected via VGA cable that was screwed on; it is quite difficult to disconnect it from the computer. And also when the monitor hits the ground, you hear light sounds of glass breaking and you can see as the monitor rolls that the glass is missing, yet none is to be seen - it's a hollow chassis. Obviously done as the stuff was shoved off in Kelsey's direction so as to avoid injury.
Factual error: There are many factors concerning Niles' post-operative state which are very inaccurate. After open heart surgery it takes at least three months for the chest bone to heal, Niles would not be able to drive a car for weeks, nor would he be able to lift or twist which he does on the show. Also, he doesn't sport a scar. Your chest bone begins very high up, just below your neck, his scar should have been visible on numerous occasions.
Continuity mistake: Niles is at Frasier's apartment, visiting with Marty. He takes some crackers out of his black fanny pack and puts the fanny pack behind him. We see it there when he stands up but there is a shot where there is nothing on the couch behind him. It then goes back to showing the fanny pack.
Continuity mistake: This is a mistake within a mistake: Niles and Frazier are trying to get the wrapped presents out of F.'s apartment and replace them with others, to play a trick on their dad. They are getting completely mixed up, and presents are being taken out and then returned by mistake, mostly by Frasier. At one point Niles has a pile of gifts which he puts down on the table behind the couch - the shot shows him holding them, but the next shot shows a completely different pile of presents, even though it is supposed to be the same pile. Frasier picks this pile up. It soon changes again in the next shots.