Continuity mistake: At the end of the episode, the Mercedes S class that is burning is not the same car that Castle drove previously in the episode, it's an older model one. Either it's a mistake or has to do something with the plot resolution, decide yourselves...
In the Belly of the Beast - S6-E17
Continuity mistake: After Beckett gets back from her assassination assignment, she washes up in a bathroom and places the gun she was given in her front waistband near her stomach. Then she goes to meet Lazarus (Vulcan Simmons) in the basement, who realizes she's Beckett. When she pulls out the gun, she reaches behind her back to get it. (00:27:25 - 00:31:00)
Continuity mistake: At the end, Beckett spills coffee on the letter. In the first shot, before she frantically blots the spilled coffee, it says that the writer is "a junior at P.S. 370." Afterwards, this has changed to "P.S. 311." (00:40:40)
Like Father, Like Daughter - S6-E7
Continuity mistake: When Castle and Alexis visit Frank at the prison in Pennsylvania, as he's brought in, the guard removes his handcuffs. In one shot, Frank's arms are flat on the table, and in the following shot, his left arm is raised. (00:08:20)
Continuity mistake: When Kate gets a call from Lanie who asks her to come down to the morgue, Kate's jacket is beige. When Kate enters the morgue, it's black/dark blue. (00:07:40)
The Good, the Bad & the Baby - S6-E10
Continuity mistake: When Lanie arrives at the precinct to check on the baby, her shirt is tucked into her jeans, as seen when she bends over to pick the baby up. When the shot changes to behind Lanie, her shirt is untucked, and then when the shot changes to the front of Lanie, her shirt is tucked in again.
Continuity mistake: In the flashback of Beckett and Montgomery, when Montgomery finds her in the archive room studying her mother's murder case, Beckett says that she was unable to find any leads worth following. But in season three, when Montgomery explains to Castle that that encounter was when he first met Kate, he says "she found some things that didn't add up", suggesting that she did find a clue worth investigating.
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.