But that rarely happens. Senior officials at the Bureau of Alcohol, Tobacco, Firearms and Explosives regularly overrule their own inspectors, allowing gun dealers who fail inspections to keep their licenses even after they were previously warned to follow the rules, according to interviews with more than half a dozen current and former law enforcement officials and a review of more than 100 inspection reports.

One store was cited for failing to conduct background checks before selling a gun. Another store owner told investigators he actively tried to circumvent gun laws. One threatened an A.T.F. officer, and another sold a gun to a customer who identified as a felon. All were previously cited by the A.T.F. In each instance, supervisors downgraded recommendations that the stores’ licenses be revoked and instead let them stay open.