Revealing mistake: Frasier comes in complaining how workmen in the service elevator have dribbled chili dog on his foot. He is holding out his shoe saying "How ever will I get that stain out", and Eddie runs over and starts licking what we think is the shoe. Frasier makes a comment about dog saliva being the universal solvent. In fact, Eddie does not even touch the shoe, but licks the floor in front of Frasier. Obviously the floor was treated with something Eddie would enjoy licking up.
Revealing mistake: Near the end, in Cafe Nervosa, the guy that wants to sue Frasier gives Niles a push. Niles stumbles backwards, knocking over some chairs and the coat rack before falling down, taking the table by the window with him. If you look closely just before he hits the table, the woman sitting there (rather than moving out of the way), very subtly pulls on the table, so as to upend it so Niles can go down with it.