Continuity mistake: When Beckett, Castle and Capt. Montgomery are walking through the precinct, after Castle and Beckett checked on Alexis as she had been categorizing personal belongings, and Castle is talking about how he watched Alexis from the bushes the first day of preschool; Montgomery states he followed his son's bus all the way to the Adirondacks. In the final episode of season 3, Montgomery had his two daughters, and wife at his funeral. There was no son.
Factual error: In the beginning of the episode, the neighbors do not hear shots fired because the murderer uses a homemade silencer. At the end, when they find the murder weapon in the dumpster at the murderer's apartment, it is a revolver. Silencers (homemade or otherwise) do not work on revolvers because the gas escapes from the sides, not exclusively the end of the barrel.
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.