The accounts are run for the Post Office by the US bank J P Morgan
The Post Office has said that 25,000 pensioners mistakenly overpaid their benefits in the summer can keep the money.
At the end of August, around 150,000 people using the new electronic "Direct Payments" system were paid twice what they should have been.
When the error was spotted, the money was reclaimed from most of the accounts. But 25,000 people had already made withdrawals and did not have sufficient funds left to repay.
After lengthy discussions with its suppliers, the Post Office has confirmed it will not ask for the money back, but has told pensioners they can make voluntary repayments if they wish.
It is writing to all customers who received double payments to explain the situation.
The blunder was deeply embarrassing for the government which had been trying to move everyone onto the new system which pays benefits direct into bank accounts.
The affected people had already switched from order books - which are being phased out - to the new Post Office Card Account.
Pension payments pass through the US Bank JPMorgan, in New York.
A clerk there noticed the double payments and turned the system off. As a result, up to 200,000 pensioners were unable to gain access to their money for several hours.
POST OFFICE HELPLINE
Pensioners with any concerns can call the Post Office banking helpline on 08457 22 33 44 (or text phone 08457 22 33 55).
At the time, DWP Minister Chris Pond told BBC Radio 4's Money Box:
"It is quite unacceptable that pensioners were left in this position. I have had an apology from the Post Office and I have had an urgent discussion with the chief executive."
And on the double payments, he said: "We do not expect the Post Office or its suppliers to be pursuing pensioners for this money."
The blunder led to top level talks between the government and the Post Office, after which a PO spokesman said:
"We have talked with the minister and understand what he has requested for the future.
"We will be working with our suppliers to meet his request so that we can avoid any similar problems happening again."