An overwhelmed testing site in Connecticut is forced to close after a gun threat.

BRISTOL, Conn. — Crowds grew increasingly frustrated and unruly as wait times stretched for hours at a coronavirus testing site here that is run by Genesys Diagnostics, an Oakdale-based company. Some were getting out of their cars to ask how much longer it would take — others were arguing with the staff.

“We opened at 12, and when we came, this place was already flooded,” said Xavier Quintana, a specimen collector who estimated that there were close to a thousand cars packed into two parking lots.

People were growing panicked, Mr. Quintana said, and the staff could not keep up with the demand. Only a few people were working out of a small trailer set up in one parking lot.

Then, a woman who said she was waiting in line called the company shortly before 1:30 p.m. Tuesday, threatening to shoot people at the site.

It was shut down within minutes, forcing many to go home without a test. The police did not find a gun or a credible threat on the scene but were investigating the episode, said Lt. Geoffrey Lund, the public information officer for the Bristol Police Department.

On Wednesday morning, a site in nearby New Britain that is also run by Genesys Diagnostics shut down after people became agitated, the site’s manager, Aaron Williams, said.

“People are getting fatigued and getting mad and causing problems,” Mr. Williams said.

Staffing shortages have been an issue at numerous testing sites, he said, especially in areas where employees are being asked to work outside in cold weather.

At the Bristol site, the size of the staff was increased to about eight employees the next day, and police cars were set up around the site to limit the size of the crowd to about 100 cars at a time.

José Mamguia, a 22-year-old resident of Bristol who was turned away on Tuesday after the shooting threat, nodded off in his car on Wednesday after about two hours of waiting in line.

By 1:30 p.m., some in line had already been waiting for over four hours. Erin Dunn, 43, of Bristol, had arrived at 9 a.m. and was growing increasingly fatigued. As a retired nurse, she understood the stress that the staff must have been under, but she was starting to feel symptoms after a probable exposure to the virus.

“I could go to sleep right now,” she said.

What she wanted was more information. Nobody had told her how long the wait would take or what the process would be. Messages left for Genesys Diagnostics on Wednesday were not returned.

Grace Bianchi, 63, had already been to the New Britain site earlier in the morning. People there were getting mad and driving through barriers, she said, and she and her husband were told to leave and go to the Bristol site instead.

“I didn’t know I had to have lunch and dinner in my car,” Mrs. Bianchi said as she scanned seven other rows of cars in the parking lot. She needed to get a test to return to work next week.

“This is ridiculous,” she said. “Working people should go first.”

She and her husband didn’t get tested until 3:30 p.m.

There is no evidence that protesters and threats have slowed down the country’s mass vaccination campaign. A spokesman for the National Sheriffs’ Association, Patrick Royal, said on Wednesday that his group had not heard of widespread instances of violence at testing sites across the country.

Adeel Hassan contributed reporting.