In its Bronx investigation, the center substantiated allegations of physical abuse or neglect against 13 employees, but the state failed to persuade arbitrators to fire any of them. The center and the Bronx district attorney each said recently that they lacked sufficient evidence to bring criminal charges. Ten of the employees still work for the state.

“Cases involving individuals with special needs are particularly challenging,” the center said in a statement. “Victims may be nonverbal or unable to provide testimony, and witnesses may not be cooperative.”

In the Bronx and statewide, arbitration between the state and the Civil Service Employees Association can allow abusive employees to keep their jobs. One arbitrator returned an employee to work “over strong objection of state,” records showed, after he grabbed a resident by the pant leg and shirt and shoved the resident into a recliner with enough force that it hit a wall.

Mark Kotzin, a C.S.E.A. spokesman, declined to comment on specific cases. He said the employees have “some of the hardest work imaginable,” and when they hear of abuse or neglect, “they get upset and want the abusers gone.”

While many employees are compassionate, some boil over, case files reveal.

“I will pop you like a zit. I [expletive] hate you,” one worker said to a resident. “Why do you act like such a crazy person?” another worker asked a resident. A third proclaimed that she “did not give a [expletive] about the consumers” — a term the state uses to refer to developmentally disabled residents.

“Most families have no clue what they’re putting their kids into,” said Mr. Carey, the advocate, adding, “It’s all hidden from the general public.”

In some places, the police appear to be more frequently intervening, pa rticularly at Sunmount in northern New York, one of the last large housing facilities. One worker, for example, faced nine disciplinary charges after pouring Gatorade on a resident, slapping and jumping on a resident, and laughing while two colleagues were hitting a resident. He was arrested in 2015 and resigned.