To build a rebel army, the Pentagon asked Syrian commanders last winter to nominate their best fighters. U.S. military officers spent more than a month checking each one for criminal or terrorist connections. Those who made the cut were sent to screening centers where they were questioned by American, Jordanian and Turkish officers. Then they waited, sometimes for days.

Fighters who made it to the screening centers were confused about the mission. When they learned what it was, many left. Others were found unfit, including...