The money is owed by absent parents who have been referred to the Child Support Agency.
Accounts prepared for Parliament by the Department for Work and Pensions (DWP) class £2.9bn of the £3.9bn in maintenance arrears as "uncollectable".
Ministers said the focus was on debts for children still being brought up, but old debts would not be "forgotten".
The Child Support Agency (CSA) was set up 21 years ago with the aim of ensuring that if a relationship broke down, a parent could not just walk off and leave an ex to pay the costs of bringing up their children.
The CSA's role was to calculate maintenance - and ensure that non-resident parents paid up.
But it has been dogged with problems. Computer systems did not work properly. Mistakes were made with assessments and absent parents were not tracked down.
In 2012, a new child maintenance system was introduced in an effort to draw a line under the CSA's difficulties.
The Child Maintenance Service, as the new system is called, does not consider recovering the historic debts on the CSA's books as a priority.
Around 1.3 million single parents are owed money the CSA ruled they were entitled to - and some have been waiting years.
One single parent, Marianne Miles from London, said she badly needed the money the CSA ordered her ex-partner to pay towards their son.
But after nearly seven years of waiting for payment, she has all but given up.
"I am not hopeful of seeing any of that money to be perfectly honest," she said.
"They [the CSA] haven't made much effort to get anything - even £30 a week - so I have practically written off all of it."
In 2008, Marianne Miles, from London, was made redundant from her job as a publicist. Raising her teenage son alone became a severe financial struggle.
She turned to the CSA, which ordered her ex-partner to contribute. To this day, she still has not received the money owed, which has built up to a total of £12,000.
"I try not to get angry about it, but it's not fair," she said. "If he owed the government council tax money, or for a parking ticket, they would pursue it and make sure it was paid. But when it's paying maintenance towards your own child, it is not deemed as important."
Her son is now 20 and has left home, but Marianne is still responsible for him financially.
"Just because he didn't have it before doesn't mean he doesn't need it now," she said. "It would make a huge difference.
"He's a student so it could go towards his books and his food, that I have to support, and he's learning how to drive. I'm the only person around to help him."
Janet Allbeson, from the single-parent charity Gingerbread, believes the government should be recovering unpaid maintenance, even in cases that have been going on for years.
She said: "If they do manage to get the money, even if it is years late, it can really make a difference to a child's life.
"It allows, for instance, to help with the cost of going to college, perhaps paying off a student loan, or helping a child learn to drive.
"Children in some ways get more expensive as they get older and that money can make a real difference to a child's future."
The government said it was prioritising cases where a child was still being brought up.
Steve Webb, the minister with responsibility for child maintenance, said: "We are quite clear that just because money has been owed for a long time it does not mean it should be forgotten about. It should be paid.
"Of course everything cannot be a priority. We have been quite transparent… and we have made it clear that where there is a child still at school or being brought up - that is the first priority.
"But we are not writing off or ignoring old debt."
But figures in DWP accounts published in December raise questions about how much of the arrears backlog is likely to be recovered.
The £2.9bn listed as "uncollectable" relates to cases where either no payment has been received from the absent parent in the previous six months, or no arrears schedule has been put in place.
The DWP said the definition of "uncollectable" in the accounts did not give a true picture of the level of debt that is beyond its enforcement powers - and in fact only £244m was permanently "uncollectable".
This report will be broadcast on Sunday, 25 January 2015 at 11:00 GMT on BBC Radio 5 live.
You can listen to other editions of 5 live Investigates on the BBC iPlayer.
You can contact 5 live Investigates by emailing goldberg@bbc.co.uk or comment on this story to @BBC5Live on Twitter.
Billions of pounds owed in child maintenance may never be recovered, government accounts show.