Continuity mistake: When they first arrest Penelope they show her first fingernail broken. In the next scene when she's talking about the nail, she shows her middle fingernail broken.
Continuity mistake: When Beckett and Castle kiss at his party at the end of the episode, the camera is behind Castle, and Beckett's head is leaning toward her left shoulder. When the shot switches to behind Beckett, her head is leaning to the right. Then it switches back to left. (00:38:05)
The Fast and the Furriest - S5-E20
Continuity mistake: Beckett's sweatshirt changes position on her left shoulder throughout the scene in which she and Castle are sitting in bed discussing Bigfoot. Her sweatshirt is initially covering her shoulder and then keeps going lower and lower off her shoulder every time the camera comes back to her .
Revealing mistake: After Castle and Beckett are arrested for interfering in Chief Brady's investigation, a deputy tells them that Brady wants to speak to them; Beckett stands up to leave the holding cell, but is held back by the pair of handcuffs chaining her to the bench. The cuff is not actually on Stana Katic's wrist; she's just holding it in her hand.
Factual error: On the way to the Hamptons, Castle's car is driving on the left side of the road, even though they're in the US (where people drive on the right side of the road). (00:03:10)
Continuity mistake: Castle and Beckett are talking about the case in the precinct and his coffee runs out. Beckett offers to go refill the mug. She grabs his white mug in her right hand. In the next shot, it is now in her left hand. A blue mug is in her right hand instead. (00:05:15)
Character mistake: The killer's motive for the murder is getting kicked out of the band akin to Beatles drummer Pete Best, and Castle whispers to the camera that bass player Stu Sutcliffe would have been a more appropriate comparison. Sutcliffe quietly quit the Beatles voluntarily, while Best was fired, with his dismissal becoming widely-known public knowledge, so the comparison to Best actually is more applicable.
Factual error: The team is discussing the location of the Widmark hotel which the detective says is downtown. Beckett then looks at the map and locates it only 5 blocks from E 72nd street, which is UPTOWN in NYC. Furthermore the map shows E 72 intersecting Broadway which is on the west side of Manhattan. This entire mapping exercise is incorrect.
Other mistake: Beckett points out to the murderous security guard that some graffiti magically appears on the tape, showing the live image was replaced with older pre-recorded footage. "Do you see that? It's actually kinda hard to spot." Except the graffiti doesn't appear at the point where the live feed restarts (when the guard enters the shot), it only appears on the footage after a cut away and then back.
Continuity mistake: Castle gives Beckett a coffee mug in a very loving way by moving her hand to hold the mug by the handle while she is reading a file and not looking. Then in the very next scene she is holding the mug on the bottom in her palm. (00:05:24)
Continuity mistake: When Beckett watches the weird/creepy video and a few images after seeing the ghost at the house, she hears a little girl laugh. However when Castle watched the video, that laugh happened at the moment with the ghost at the house, not after.
Character mistake: Beckett identifies the area given for the Skype call as the whole city of Paris. It's hard to believe that Beckett knows the area of a foreign city by heart, the problem is also that she is wrong. Paris actually has an area of about 105 km² (41 sq mi).
Secret Santa - S5-E9
Other mistake: When Ryan comes into the precinct to tell everyone that the victim did not have job booked on the night of his murder, that fact is already written down (in red) on the timeline board in the background,. (00:11:00)
Cloudy with a Chance of Murder - S5-E2
Continuity mistake: When Beckett, Ryan, and Esposito are discussing Castle's date with Kristina Cottera, Beckett's fork is full of food in a long shot, which disappears in the following close-up without her eating anything. (00:32:15)
Cloudy with a Chance of Murder - S5-E2
Continuity mistake: When Beckett is trying to get dressed for work at the beginning, she takes off a blouse she deems unsuitable and the bullet scar on her chest, which is usually at the start of her cleavage line, is missing. (00:00:25)
Continuity mistake: When Castle and Beckett are theorizing about Melanie's relationship with Senator Bracken, Beckett's right hand is visible holding her cell phone. When the shot cuts, her hands are suddenly in her coat pockets, with her phone nowhere in sight. (00:06:25)
Answer: The networks are now having winter breaks. A mid season end of new episodes, when they come back after the new year, it picks up as if several months have gone by. Beckett suggested Valentine's Day as a "chick flick" to watch as she is moved in. Also some episodes are not scheduled to air in order.