More than 100 people, including 72 former New York City police officers and eight former firefighters, were charged Tuesday in one of the largest Social Security disability-fraud cases in U.S. history, which authorities said cost taxpayers as much as $400 million over two decades.

The plot, allegedly under way since at least 1988, involved four facilitators who helped coach hundreds of applicants on how to convince the Social Security Administration that they were entitled to monthly disability payments because they were unable...