Visible crew/equipment: At the hospital, after Maria tells Danny about the busboy who was arrested for the same venue and same MO, it then cuts to Jaime undercover in the store. While Jaime walks toward Marco, the dealer at the counter, the camera moves at the same pace as Jaime, and we can see the reflection of the camera on the glass (between Jaime and the camera). (00:28:20)
Continuity mistake: When Danny and Baez confront Carmine to clear a few things up, Carmine tells them to call his lawyer. Just as Carmine clicks his key fob and his car explodes, Carmine's coat is entirely unbuttoned, but when he's blown backwards and slams onto the ground, in the first overhead shot of Carmine lying on the ground, his coat is buttoned up. (00:24:20)
Continuity mistake: Danny and Jamie are at the pistol range shooting targets. The vantage point from down range shows them both shooting tight shot groups. However, when they are finished shooting, they hold up their targets; their shot groups are not as tight as the down range perspective. (00:34:01)
Continuity mistake: Danny and the ESU enter the warehouse basement to find Bobby, and when they start to return fire, the first guy they shoot knocks over the white plastic lawn chair (beside the recliner) as he goes down. Then, in the next shot facing the door, that white lawn chair is upright (not the yellow table chairs), but in the next shot it's lying on the floor again. (00:32:40)
Factual error: The badge the ATF Special Agent is displaying on his suit jacket is wrong. It's the wrong shape and style of an actual ATF Badge. (00:02:57)
Officer Down - S1-E4
Factual error: Danny and Jackie Curatola are with ESU to make an arrest at a garage. They are behind cover with their weapons drawn. Although there is no suspect shooting at them, both Danny and Jackie have their fingers on the trigger, which is against police training unless you are going to engage a subject by shooting your weapon. The ESU officer right next to them does not have his finger on the trigger. (00:38:04)
Other mistake: Danny, while driving home with his family, hits a man with his car. Danny gets out of the car to attend to the man, but his wife, Linda, who is a nurse, does not get out of the car to render aid to the man. (00:02:18)
Plot hole: Jamie and his witness are in a patrol car when they are attacked by criminals in a van firing machine guns at them. Jamie (of course) chases them away after a prolonged firefight, and when the van is driving away we see another patrol car also shooting at the fleeing suspects. There didn't appear to be any cars with Jamie, nor did anyone come to his aid during the gunfight, and this other patrol car was facing the wrong direction.
Plot hole: Early in the show, Jamie is in a patrol car escorting a witness when they are attacked by several guys in a van. The side windows of their cruiser seem shatterproof and withstand many bullets, while the rear window is shattered immediately. NYPD cruisers do not have bulletproof windows.
Answer: It's a common cop show plot device. If the police on the show always acted in the most logical way (i.e. not alerting a suspect to their presence) the show wouldn't have the opportunity for an exciting foot chase.