Revealing mistake: When Joey comes in with the shoebox containing the chick, it looks like a normal sized box, about 5 - 6 inches deep. But when Joey takes the lid off, we can see that the chick must be elevated on something inside the box, since we can see its entire body. Surely, the chick can't be more than 6 inches tall?
Other mistake: In the scene where Rachel is talking to Chandler in the hall, after she says "the 4th of July", it cuts to a shot from behind Rachel's head. If you look on the right hand side of the screen, you can see a piece of wood in the foreground. This probably came from Joey and Chandler's doorframe, through which this view would have been filmed. In addition, this door was supposed to have been closed throughout the entire duration of this scene.
Other mistake: This is the episode where a guy across from Monica's apartment is spying on them. Joey says he caught him spying into his and Chandler's apartment yesterday. This is impossible considering the layout of the building i.e. the fact that the only part of Joey and Chandler's apartment which faces toward the spying guy has the stairwell and hallway blocking it from view (not to mention the protruding part of Monica's apartment). There is another window in Joey's apartment but it faces completely the wrong direction. The bathroom has no windows. The angles just don't add up.
Plot hole: The times in this episode vary a lot. Monica and Phoebe first bet about Rachel giving birth, then it goes to the scene with Joey and Chandler at the movies. It then returns to Monica and Phoebe the next day with Monica losing the bet. However, a little while later, it goes back a day and shows Joey and Chandler at the movie premiere again.
Continuity mistake: In the scene with Robin Williams and Billy Crystal, there is a shot of Rachel, and her hair is hanging straight down. The next shot of Rachel is from a different angle and there is a section of hair behind her ear; in the shot after that it is hanging straight again, and is then behind her ears again.