Continuity mistake: When Paul the Wine Guy first comes to Monica's apartment, Phoebe is standing in the kitchen telling Rachel that she just pulled out four eyelashes. But in the very next shot, you can see her sitting in an armchair in the living room. [Broadcast version only - in the extended DVD version there's a shot showing Phoebe walking to the chair.]
Visible crew/equipment: Right after Rachel says, "Do you know that feeling when you're trying to blow a Saint Bernard out your ass?" the camera cuts to a view of Monica and Chandler. Behind Monica and Chandler, part of the set is visible, along with a man on crutches.
Plot hole: In the opening scene Phoebe and Chandler discuss how Phoebe knows Chandler is going to propose: by walking in on him whilst he was looking at a ring catalogue on the toilet. However, in the previous episode Chandler leaves the apartment right after he found out about Monica booking the wedding slot (on the answerphone) and doesn't come back until he 'freaks out' at Monica and then it is revealed that it was just put on and Phoebe is in on the plan. It doesn't add up, as Chandler would have had to have been looking at the catalogue between hearing the message and deciding to propose, in the same afternoon when he wasn't even in the apartment.
Factual error: The monkey is allowed outside his cage at an airport gate. Animals, with the exception of assistance dogs, cannot be transported in the cabin with passengers, and this rule applied in the 1990s as well. There is no way a monkey would be brought through a security checkpoint into a terminal concourse of a NYC airport and allowed aboard a regular commercial airliner.
Factual error: When Ross shows Chandler the book, he claims it is in "the biggest library on campus" at NYU, which would be Bobst library on Washington Square Park South. Later, in Central Perk, when Ross tells Rachel and Joey about the people "making babies" in front of his book, Joey says, "Wait a minute. Fifth floor, against that back wall?" Bobst's fifth floor is conference rooms only; there are no stacks on that floor. Joey might have been confused, but Ross should know better, being a professor and having just visited that location.
Factual error: Phoebe discovers that one of the babies has wandered away, and finds the baby inside the open drawer. Yeah, no. These babies aren't even walking yet, much less are they able to lift themselves up past the height of their own bodies and launch themselves into a drawer.
Plot hole: Monica is trusting Rachel to make a traditional English trifle and being a top chef (or an even averagely intelligent human being) she would know what ingredients are used in one. So there's no way she could have missed Rachel cooking beef with peas and onions during the cooking process. Rachel's cooking it right there in Monica's kitchen while everyone else is there!
Continuity mistake: When Phoebe and Joey are at the coffee house discussing their plan to ruin Ross' blind date just before Joey says, "Ooh I hear that's bad" the pendant on Phoebe's necklace is smooth and blue. When the camera pans to Phoebe after Joey says his line the pendant is flipped over and reveals a new design. This continues to happen throughout the scene.
Continuity mistake: After Joey runs out of the apartment to go make the fake videotape, when it cuts to Rachel walking back into the apartment the table with the tablecloth is quite far away from the front door, but when Caitlin knocks and Ross rushes to go flirt with her that table is near the door.