Deliberate mistake: They show footage of Max and Larabee scouting out the entrance to a bank. This same footage was used in the earlier episode "The Treasure of C.Errol Madre".
Thank You and Good Night - S1-E8
Deliberate mistake: In the Maisels' 1954 wedding reception flashback, Midge hired professional dancers who perform the "Bottle Dance" (where they slide on their knees). This famous dance, often performed at Jewish weddings and social events, was choreographed by Jerome Robbins for "Fiddler on the Roof," which debuted on Broadway in 1964. It did not exist in 1954 but, as an inside joke, while Midge watches, she says, "That's really good. They should do something with that."
The High Life / Best Friends - S2-E10
Deliberate mistake: In "The High Life", there is a scene where Stinky submerges his head in a tank of water. He delivers the line, "Gee willickers, it's still ticking", and everyone somehow understands what he is saying with his head underwater.
Suggested correction: Nobody responds to what Stinky says, so there's no reason to assume that everyone understood him.
Deliberate mistake: After it is revealed Won-hyung has taken over Yeom's body, he is shown walking alone on the street and says aloud "She's not afraid of going to Hell? I can't make it so easy for her. Will she be able to leave behind the person she's been pining for so easily? I can't wait to watch the heart-rending sight." There is no reason for him to say this out loud. He is alone and not talking to anyone. The only reason he is saying this is so that the audience knows his plan. (00:33:23)