In Israel, where there is a high level of alertness to threats and civilians are taught to act immediately if they see anything suspicious, there are multiple incidents of civilians identifying a terrorist threat before a policeman or other authority, with the threat being at least partially mitigated as a result. See e.g.:
Passengers escape Bat Yam bus shortly before bomb explodes:
Passenger David Pappo described, in a shaky voice on Sunday afternoon, how he was on the bus when he heard a young boy mention a bag sitting near him unattended.
Pappo said he alerted the bus driver, who asked whose bag it was, but received no answer.
Pappo said he then opened the bag – which he described as large and black, with two handles – and found what he said looked like a pressure cooker with a red wire coming out of it. He said he started telling people to get off the bus, and when the driver stopped people started filing off.
Terror attack 'miracle': I screamed get out! Suspicious object!
"There were three young yeshiva boys sitting around it, around 14-15 years old," Amuyal added, reconstructing Wednesday's events. "I looked at the bag and had a very strong bad feeling about it. It was new, very new, with a zipper and it seemed suspicious. At that very moment I told them 'move quickly, it's a suspicious object, evacuate the area immediately'."
Amuyal then took out his cellular phone and called the police to report the suspicious bag. "As I was on the phone I felt a huge blast that threw me back. I was about a meter and a half (4.9 feet) away from the device, I flew 4-5 meters (13-16 feet) back and my body caught fire. I tried to put out the fire with my hands; I got up, walked around 15 meters away from the scene and sat on a railing nearby."
Bomb found at central Tel Aviv intersection: 'We were lucky':
A black trash can instantly became the Yarkon District Police’s worst nightmare: A possible explosion in an area packed with pedestrians and cars. A citizen who intended to throw out his trash opened the can, and to his horror found a gas canister wrapped in black electrical tape.
The man immediately called police, who wasted no time in dispatching large numbers of police officers - mainly due to the tense security situation. An initial examination immediately revealed that this was more than just another suspicious object and could potentially explode
Mideast Violence Is Nearly Non-Stop:
Another would-be suicide bomber, at a trendy Jerusalem cafe, was thwarted when the cafe owner, a waiter and a customer jumped him, shoved him outside and grabbed his bag after they saw wires dangling from it. "Who, me?" the man asked when confronted, cafe owner Gabi Aldoratz told Israel radio.
At a shopping center in Pardes Hanna, a city in Israel's north, a resident spotted a suspicious object and called police. As a bomb disposal team approached, the bomb exploded, police said. No one was hurt.
Of course there are also many, many false alarms.
It is not clear whether the same kind of threats are present in significant numbers in the United States, but there is at least one instance of a potentially significant threat being identified by a civilian:
The 2010 Times Square car bombing attempt was an attempted terrorist attack that occurred on May 1, 2010, which was foiled when two street vendors discovered a car bomb and alerted New York City security personnel to the threat after they spotted smoke coming from a vehicle. The bomb had been ignited, but failed to explode, and was disarmed before it caused any casualties.