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.
Plot hole: When Monica falls off the chair you can see that she is wearing a white bra because you can see some of the cup and the strap (when she says "you're not getting it". Towards the end of the episode (still the same day) we find out that Monica is meant to have the same bra on as their new cleaner, Brenda. The bra she is meant to have on is 'pink with the lacy cup'. Rachel notices that Monica is wearing that bra and she shows Monica. What happened to the white bra she had on a few scenes before?
Continuity mistake: After Chandler has been playing the Ms Pacman machine all day he claims his hand has seized up and it appears to look like a claw. From then on in that scene, his hand manages to move around fine. It goes in his pocket, and he waves it around like normal, and then when he mentions his hand again at the end of the scene, it appears to have seized and is clawlike again.