A Panorama survey of hundreds of local councils across Britain discovered that 7,585 families had had their weekly housing benefit cut to this level.
Charities say some families affected could end up losing their homes.
The cuts are part of the government’s drive to get unemployed people back to work by capping their payments.
The limits are £23,000 in London and £20,000 in the rest of the country.
Benefits can only be capped if someone receives housing benefit or Universal Credit.
The amount of money above the limit is taken from housing benefit or Universal Credit.
It means some people can lose all their housing benefit, except for a nominal amount of 50p.
‘Hole in rent’
Alison Garnham, chief executive of the Child Poverty Action Group, said: “Removing people’s housing benefit basically means that people can’t afford their home, so it puts people at risk of homelessness.
“It also means that they have to use money that’s intended to buy food for their kids and for their other living expenses – this has to be used to plug the hole in their rent.”
The government says the benefit cap is fairer because it levels the playing field between families in work and those reliant on benefits.