Plot hole: There's no way Phoebe has lived to 31 without knowing her birth date. She's had "real" jobs (like the massage parlor) and those require social security numbers and accurate birth dates. She didn't go to high school, but she mentions classmates, so she must have gone to elementary/middle school, and those required birth dates even in the '70s and '80s. She's also "lived in Prague," which she couldn't have visited without a passport - which requires a birth certificate.
Revealing mistake: In the scene in the coffee house, Phoebe is talking to Monica who is drinking a cup of coffee. Towards the end of the scene, the camera is on Phoebe who is smiling & laughing at Monica. In the right hand corner is a side view of 'Monica' however, it is not Courtney Cox in the scene, but a completely different actress. [Widescreen version only, although the "wrong" nose is visible in the original]. (00:06:20)
Visible crew/equipment: After Ross and Rachel cover the table with the sheet, when Phoebe walks in Rachel says they can turn out the lights and watch the movie, and in the next shot we can see the end of the set wall, the soundstage and a crewmember's leg.
Continuity mistake: When Rachel finds out that Ross and Julie are getting a cat, watch Rachel's neck. When she says "together?" she's not wearing a necklace; cut to Ross and Julie for a fraction of a second, then when it cuts back to Rachel saying "both of you?" she's suddenly wearing a necklace. It then disappears again when she says "isn't that just lovely?" hanging off the side of the tray she's holding. (00:05:15)
Visible crew/equipment: During the scene when Phoebe breaks into the other Phoebe's house and gets caught, the older Phoebe was telling the younger one that the man in the photo is not Frank Buffay, but is Chuck Mangioni. Right after that, as Phoebe 2 walks towards the camera, you can see that the door behind her moves open, and there is a visible shadow of someone who walks by that doorway. (00:19:00)
Visible crew/equipment: When Ross shows up at Chandler's office and they both head out to go to the Hard Rock Cafe, just as Chandler walks out the door we can see the edge of the right wall set, allowing us to see the hallway Chandler just walked into and even Chandler's arm.
Visible crew/equipment: When Joey's in the kitchen holding a football waiting for Janine to come home, so he can talk to her about getting rid of the girly stuff, just as Janine walks into the apartment the camera can be seen at the right side of the screen.
Visible crew/equipment: While Phoebe and Rachel are with Monica in the guestroom, Phoebe decides that Rachel should be the one to get the nice room, and right before Monica tells Rachel that she's the first guest at "Hotel Monica" we can see the end of the bedroom set wall, with the kitchen and living room visible.
Visible crew/equipment: After Janine tells Joey that she'll apologize to Monica, in the shot of Janine walking into the hallway the end of the hall's set wall can be seen at the left side of the screen, and the corner of the camera can be seen moving at the right side of the screen.
Other mistake: When Phoebe and Ross are at JFK, some of the people who are at that airport are the same people who are also at Newark. It's particularly funny to see the people behind Ross and Phoebe at JFK when Phoebe claims they're on their honeymoon, and when it cuts to Rachel at Newark as she's looking for her boarding pass, those same people are there too.
Visible crew/equipment: When Joey's telling Phoebe about having been at a nice restaurant with his date and how she took fries off his plate, in the shots facing Joey in the restaurant we can see the end of the set wall, with a 2x4 framed plywood board and a heavy duty extension cord hanging there.
Visible crew/equipment: After Ross goes in for his disastrous tan, when it cuts to Joey and Rachel walking into their apartment the hole (used for a camera angle) in the wall can be seen at the right side of the screen. As an aside, when Ross leaves Monica and Chandler's apartment later just as Rachel walks in, that hole is seen in the hallway and it's covered up.
Continuity mistake: When Joey is counting the floors of Ross' building (the first time) to find out where Hot Girl lives, he is wearing a black top with a zip. When he rushes over to Ross' building he walks down the hall and knocks on what he believes to be Hot Girl's door. In these shots he is wearing a dark purple button up shirt. As soon as Ross answers the door, Joey is wearing the black top with a zip again. (00:03:15)
Continuity mistake: In season 1 ("The One with George Stephanopoulos") the date's October 20th, and Ross says his birthday "was 7 months ago". In season 4 ("The One With Joey's New Girlfriend") Gunther is pretending to collect birthdays, and Ross starts saying "mine's December the..." before being cut off. In season 9 ("The One Where Emma Cries") Ross says his is on October 18th. (00:15:10)
Visible crew/equipment: When Phoebe bends over to tie her shoelace before the first run, and later when she surprises Rachel at the park while running, the mic pack bulge under Phoebe's top can be seen on her back, and when Rachel bends over there's a bulge at her back as well. It's also a bit noticeable the third time they run.
Visible crew/equipment: When Phoebe tells Chandler that Monica wants to make up and that he needs to find her, when Chandler leaves the room we can see the end of the set wall, with the hallway and another hotel room door, at the left side of the screen.