The paths to violence for the United States-born attackers varied. Some were recent converts to Islam. At least three who were born in the U.S. had previous criminal histories, and one had a history of mental illness. One seemed to have radicalized after spending time in Yemen. Another became radicalized after being convicted of lying to F.B.I. agents — denying he had made plans to travel to Somalia when in fact he had.

Security experts argue that the risks of routine travel — including the U.S. visa waiver program, which allows citizens of Britain, France, Belgium and 35 other countries to enter the United States without a visa for stays of up to 90 days — are greater than the threat of foreign terrorists coming through the refugee program.

“Further restricting the acceptance of refugees does not address the most likely vulnerability to attacks from abroad, which is the large number of people from visa-waiver countries involved in the conflict in Syria,” said David Sterman, a researcher for the International Security Program at the New America think tank who has been cataloging terrorist attacks carried out since Sept. 11.

Nidal Malik Hasan, who killed 13 people in a mass shooting at Fort Hood, Tex., in 2009, was born and raised in Virginia. Mr. Hasan had exchanged messages with Anwar al-Awlaki, an American radical cleric who was later killed by a drone strike in Yemen. Despite those exchanges, investigators have not linked Mr. Hasan’s attack to terrorism.

Dzhokhar and Tamerlan Tsarnaev, the brothers responsible for the Boston Marathon bombings in 2013, settled in the United States after their parents were granted political asylum, which involves a less extensive vetting process than the program for Syrian and Iraqi refugees.

At the time of the Boston bombings, which killed three people and injured more than 260, Dzhokhar Tsarnaev was a naturalized American citizen, and Tamerlan had a green card.

Since 2001, “hardly any foreign-born have committed (or tried to commit) terrorism in (or on the way to) the U.S.,” John Mueller, a political scientist at Ohio State and the Cato Institute who tracks terrorism in the United States, wrote in an email.

Richard C. Reid, who tried to detonate explosives in his shoes on a flight from Paris to Miami in 2001, is a British citizen, and would not have needed a visa to enter the United States. Umar Farouk Abdulmutallab, a Nigerian man who tried to detonate explosives in his underwear during a flight from Amsterdam to Detroit in 2009, had a tourist visa.