In a scathing report, the Republican-led House Oversight and Government Reform Committee said New York overcharged taxpayers by $15 billion since 1990.

In 2011, New York charged a per-diem rate of $5,118 for residents of the institutions, a network of 11 centers that now house about 1,300 people with severe developmental disabilities. Over the course of a year, Medicaid spends $1.9 million for every resident, or $2.5 billion in total—with half coming from the federal government. But the cost of running the institutions is only a quarter of that amount.

The report said New York took advantage of a complex formula and kept federal officials in the dark for years. It also faulted the federal agency that oversees Medicaid for waiting years before investigating after becoming aware of the high payments.

“This is intentional fraud,” said Arizona Rep. Paul Gosar, a Republican committee member.

The committee’s report said Gov. Andrew Cuomo’s administration refused to cooperate with the investigation. Joshua Vlasto, a spokesman for Mr. Cuomo, said the report’s conclusions were “wrong and totally misleading” and that a threatened “precipitous reduction” in funding would jeopardize administration efforts to modernize and restructure its Medicaid program.

The report could pose budget problems for Mr. Cuomo. Republican lawmakers in Washington are putting pressure on the federal Centers for Medicare & Medicaid Services to crack down on New York’s reimbursements. CMS officials had indicated that they favored a gradual reduction in the rates over several years.

But at a Thursday hearing, Penny Thompson, a CMS deputy director, suggested that the agency may take a tougher line. “You can expect to see a rate that’s about one-fifth of its current level,” Ms. Thompson said, without specifying a time frame. Such a reduction would reduce the annual federal reimbursement by about $1 billion, punching a hole in New York’s $54 billion Medicaid program.

The skewed methodology traces back more than 20 years, when New York got permission from the federal government to use a different formula for state-run developmental centers, assuring officials that the rates would hew close to costs.

But almost immediately, reimbursements began to skyrocket. The new methodology allowed New York to bill Medicaid for ghost patients: When a patient was discharged from a state-run facility, New York retained nearly two-thirds of the reimbursement amount. The formula also double-billed taxpayers: Many of those patients who left the centers moved into Medicaid-financed group homes.

Between 1990 and 2011, the daily reimbursement rate grew to $5,118 from $348. Ms. Thompson said it wasn’t clear if CMS “completely understood” the cost projections when it approved the rates. CMS officials acknowledge they first became aware of the problem in 2007 but waited three years before launching a probe.