Lawyers for the families, who are all from the London area, say the policy is "cruel and arbitrary" and reminiscent of the days when the only option for out-of-work families was to send their children to the workhouse.
The lawyers warn the policy threatens to reduce the income of poor families to a level that makes it impossible to provide adequate food, clothing and other essentials.
They say two of the test case families will be expected to survive on sums which are less than that afforded to asylum seekers and which, if they were asylum seekers, would be considered 'destitution' level.
The application for judicial review, expected to last three days, is before Lord Justice Elias and Mr Justice Bean at London's High Court.
A legal team acting on behalf of Iain Duncan Smith, Secretary of State for Work and Pensions, is defending the policy.
The capping affects housing benefit, child benefit and child tax credit to families who do not work sufficient hours to qualify for working tax credit and is set at £500 per week for couples or lone parents.
Hopkin Murray Beskine solicitors, acting for the families, say the impact of the cap is most acute in London because of the lack of affordable housing in the capital and high housing benefit costs.
L arger families with three or more children are particularly badly affected and the policy discriminates against ethnic groups with large families, including Orthodox Jews, Catholics and Roma, say the solicitors.
S urvivors of domestic violence, who often face particularly high housing costs, are also victims.
Two of the claimant families live in private rented accommodation which is already too small for them but from which they have no real option to move.
Many of the worst affected families live in temporary accommodation provided to them when they were made homeless, say the lawyers.
Such accommodation is frequently expensive, despite being of poor quality and frequently overcrowded.
The court will be told that one of the claimant families has been in damp, rodent infested temporary accommodation for five years.
The judges are being asked to rule that the cap breaches the UN Convention on the Rights of the Child and the European Convention on Human Rights, which protects the right to respect for home and family life.
Rebekah Carrier, solicitor acting for the claimants, who come from the boroughs of Hackney, Haringey, and Hammersmith & Fulham, said: "This is a cruel and arbitrary policy.
"It will have a catastrophic impact on our clients and many thousands more vulnerable families. They face street homelessness and starvation.
"The Government suggests that this policy is about making the system fairer, so that those who are not in work do not get more than working families earn.
"But the comparison relied on hides the fact that equivalent families in work remain entitled to benefits including child tax credit and child benefit. Child benefit is being removed from the very poorest families who need it most.
"My adult clients did not choose to become homeless, or to live in private rented accommodation which is too small for them but all that they could afford.
"The children have no choice about the circumstances in which they find themselves, and are facing life in the sort of poverty which is reminiscent of the time when the only option for out of work families was to send their children to the workhouse."