Two-thirds - 66% - of social sector tenants affected by benefit cuts for those with extra bedrooms were behind with rent after six months, a National Housing Federation survey suggests.
And it said 38% were in debt because of the "unfair, unworkable" policy change - dubbed the "bedroom tax" by critics.
Research firm Ipsos-Mori surveyed 183 housing associations in England.
The government said it was "determined to support those who might need extra help through these necessary reforms".
Since last April, people deemed to have one spare bedroom have had their housing benefit reduced by 14%, while those with two or more spare bedrooms have seen reductions of 25%.
Critics said vulnerable and disabled people would be forced out of their homes.
But the government argued the measure would help control the billions spent on housing benefit and free larger properties for those who needed them the most.
The National Housing Federation (NHF), which represents housing associations, said the changes were "heaping misery and hardship on already struggling families, pushing them into arrears".
Chief executive David Orr said: "Now many are at risk of being evicted because they simply can't find the extra money to pay their rent. These people have done nothing wrong."
He said the government had "suddenly changed the rules and given them a false choice - move to a smaller home or pay.
"Yet we know there aren't enough smaller homes in England for these families to move into."
The NHF said the equivalent of 72,000 housing association tenants were in rent arrears because of the policy.
It said that, by last October, 15% of household affected by the cuts had received letters warning them they were in danger of being evicted.
And housing associations with affected tenants each spent an average of £73,250 before last April on measures such as welfare and financial advice services to mitigate the effect of the changes.
This would rise to an extra £109,000 on average by next month, the NHF estimated.
And it said that, according to its own research conducted since December, demand for discretionary housing payments - described by the government as providing "extra money when your council decides that you need extra help to meet your housing costs" - had tripled.
A spokesperson for the Department for Work and Pensions (DWP) said: "We have tripled the extra funding given to councils this year to £190m - some of which is specifically targeted at disabled people - and have announced that £165m will be available for councils next year to help vulnerable tenants.
"There have been many scare stories about councils running out of funding when, in fact, only a quarter of local authorities across the country made a bid for the £20m funding available to top up their discretionary housing payment allocation, and a majority of councils spent less than half of their extra funding in the first half of the financial year."
Meanwhile, Labour has accused the government of understating the number of council tenants who have had their payment wrongly docked under the policy.
Work and Pensions Secretary Iain Duncan Smith previously told MPs that up to 5,000 claimants affected should still have been entitled to have housing benefit calculated according to longstanding rules, despite the new regulations.
People who have been claiming housing benefit for the same property since before 1996 are thought to fall into this bracket.
Labour said data for local councils showed that at least 16,000 households in the UK had wrongly had benefits cut, and that the true figure could be closer to 50,000.
The DWP said regulations were being amended and it stood by its earlier estimate of "around 5,000".