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)
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.
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.