Americans are flooding the government with appeals to have their student loans forgiven on the grounds that schools deceived them with false promises of a well-paying career—part of a growing protest against years of surging college costs.

In the past six months, more than 7,500 borrowers owing $164 million have applied to have their student debt expunged under an obscure federal law that had been applied only in three instances before last year. The law forgives debt for borrowers who prove their schools used illegal tactics...