A growing number of Afghan interpreters who worked alongside American troops are being denied U.S. visas allotted by Congress because the State Department says there is no serious threat against their lives.

But the interpreters, many of whom served in Taliban havens for years, say U.S. officials are drastically underestimating the danger they face. Immigration attorneys and Afghan interpreters say the denials are occurring just as concerns about Taliban retribution are mounting due to the withdrawal of U.S. forces....