In April, in the fifth round of the NFL draft, the Miami Dolphins selected Caleb Sturgis, a kicker from Florida. He signed with the team for an annual salary of $406,000.

To the larger world, this was a routine and uneventful transaction. But in interviews with current and former NFL players, player agents and team executives familiar with the inner workings of the Dolphins, the news hit team members like a bomb.

The...