On a Tuesday night last October in Olympia, Wash., 911 operator Jennifer Rodgers stared at the list of incoming calls on her screen.

Normally, one or two calls at a time would trickle in at this hour. At 9:28 p.m., they began stacking up by the dozens like lines on an Excel spreadsheet.

An alarm alerting operators to unanswered 911 calls filled the room. It almost never sounds more than once. Tonight, it was going off constantly.

Ms. Rodgers had no idea what was happening. People in Olympia, a city of about 50,000 an hour’s drive south of Seattle, and the surrounding county were dialing 911 and hanging up before their calls were answered. Then they were dialing 911 again.

After about 15 minutes, a girl stayed on the phone long enough for Ms. Rodgers, a 911 operator for 15 years, to say through her headset: “Don’t hang up! Don’t hang up!”