HM Revenue and Customs (HMRC) has apologised to 8,000 tax credit claimants for sending their bank details to other claimants.
HMRC has apologised to 42,000 claimants for its mistake
The department said the mistake was due to a faulty printer which had been used to print 42,000 tax credit award letters, all with some sort of error.
Many of the people whose bank details were exposed live in Northern Ireland.
HMRC has written to their banks warning them to be on the lookout for any evidence of identity theft.
The department said the letters that carried other claimants' bank details did not, however, include their addresses or National Insurance numbers.
"Our understanding is that the information should not be enough to perpetrate fraud," said an HMRC spokeswoman.
The mistake came to light in the last week of April. The claimants and their banks were contacted last week.
Correct award notices are now going out to all those who received ones with errors.
"We have written to all claimants affected to apologise and to advise them what we are doing to protect them," said the spokeswoman.