I remember a great example of cutaway scenes happened in "Silence of the Lambs". The killer's last known address was discovered, and an FBI team disguised as delivery men was dispatched to that house. Meanwhile, a rookie agent rings the doorbell of the house where the killer is actually living. The scene just back and forth between the two houses, which makes it appear that the FBI team is going to find him, but they don't. The cutaways end with the killer opening the door and confronting the rookie agent.