Plot hole: In TOW Ross tells Rachel he cannot see her following a promise to Emily, the rest of the friends go into Monica's bedroom. Joey hauls out a box from under the bed, saying that he's been hiding it there since Ross said the wrong name at the wedding. Surely Monica, with her obsession for keeping the place clean, would have noticed a box under her bed?
Continuity mistake: In this episode, Rachel tells Chandler that she thought he was gay until he spent Phoebe's entire birthday party talking to her breasts. But at this point since Rachel joined the group, Phoebe hasn't had a birthday yet; that occurs some episodes later, in "The One With Two Parts."
Continuity mistake: Just after Rachel throws a bread roll at Joey, he turns to Chandler to speak to him. Chandler is holding his hands out in front of him. In the next shot, his right hand is touching his face, then in the next shot, it is lowered again.
Plot hole: Chandler has always loaned Joey money for various things (rent, acting classes, Christmas gifts, etc), so he shouldn't be surprised to learn that Joey doesn't have money for fancy dinners or birthday celebrations. Plus, Joey never seems to have a problem taking money/handouts from Chandler - why now? Also, Chandler explains that they're buying Ross a cake ($20), but why doesn't Monica (the chef and his sister) just make him one?
Continuity mistake: In the coffee house when Rachel is talking to Ross about trust, a couple comes in behind her and stands at the bar. In the next shot of Rachel they have switched positions; then they are gone, and you never see them walk past Chandler.
Plot hole: This is actually for both parts of "The One in Barbados": it's shown that Monica's hair gets all frizzy from the humidity. And yet she's shown all throughout with her hair out and loose in all its frizzy glory, though it bugs her. I could maybe buy that not one of the women have hair gel with her (not bloody likely.) or that the fancy hotel doesn't sell hair gel (possible). But resourceful Monica would have thought of something to calm down her hair, or she would have just tied it up in a ponytail or bun - she does care a lot about her appearance. It makes no sense other than for comic value.
Other mistake: When Ross is helping choose a dress and shoes for Nana to wear, he is behind the chest of drawers in the cupboard. He looks in a shoebox and finds lots of pink sweet'n low packets, then pulls down another box and lots of packets fall down on him. However, it is obvious that someone is above him showering the packets down, as they are coming from directly above him, not from on top of the door.