Continuity mistake: During the countdown to midnight, just as everyone says, "Happy New Year!", the camera cuts to Phoebe and Ross. In the background Joey and Rachel open their arms to hug each other; the camera cuts to just Joey and Rachel and they open their arms again.
Audio problem: As the nurse comes back from wrapping the new baby, we hear her say "Here she is," yet her mouth doesn't move to the words she is apparently saying. Even though her mouth is only visible during the "she is", she's not saying the words. Then she mouths some sort of "oh", but nothing is heard.
Other mistake: When Phoebe accidentally allows Chandler's ring to be sold, she tells him that she overheard which restaurant the guy said he was taking his girlfriend to, so they can go find him and persuade him to exchange rings. However, in the previous scene we see that Phoebe was paying no attention at all to that guy - she was too busy trying on jewellery. She wasn't even within earshot of the guy, and we see from her reaction that she hadn't even noticed him at all before he left the store with the ring (during which time he doesn't say anything at all).
Continuity mistake: At the beginning, when Rachel first arrives at the hospital, she is talking about the hospital not being out of rooms. She points to herself with her right hand saying 'What if I was the president?' Then when the shot changes to include Ross she is pointing to herself with her left hand, then it cuts back to just Rachel and she is using her right hand again.
Continuity mistake: In the scene where Chandler and Phoebe come into the coffee house from Phoebe's first day as his secretary, Phoebe makes no discernible effort to order - but after Chandler says it pays to know the man who wears his shoes the shot cuts to her holding a large latte cup which was not visible earlier in the scene.
Continuity mistake: In the scene where Monica and Ross discuss how Emily has decided to call off the wedding after Monica suggests they postpone the wedding, Monica is wearing a black watch on her left wrist. This disappears and re-appears a number of times throughout the scene.
Continuity mistake: The gang enters Monica's apartment and it is full of dead Christmas trees. After everybody has entered the apartment we see that the apartment door is shut; the shot then changes to the gang around the sofa, but in the next shot when Rachel answers the phone we can see behind her that the door is open.
Continuity mistake: The first time Phoebe and Monica are making bets on when Rachel will give birth, Phoebe has a bottle of water in her hand and is holding it near her stomach. She puts it to her mouth to drink some, but in the next shot, the bottle is near her stomach again.