Julianne Stafford, a special education teacher in Massachusetts, recently got a terrifying phone call: someone claiming to be a Social Security official said a number of bank accounts opened in her name in El Paso, Texas, had been linked to drug trafficking — and that Boston police had a warrant for her arrest.

The only way to clear her name, the caller said, was to pay -- by buying two $500 Google Play gift cards and giving the activation codes over the phone...