A few years back my parents realised that they had been overpaid Child Tax Credit and informed HMRC. The error was eventually tracked down to an underestimate of my Mum's salary. For whatever reason the previous year's figure, and that year's one, had been lost, so HMRC decided to enter a "reasonable guess" for what my Mum's annual salary actually was. Their guess? £1. Yes, really. One single, solitary, pound earned in a whole year. Right... I think the terms of the repayment amount to their clawing back all the money they accidentally overpaid us buy, plus a bit more on top, although I think Mum and Dad put the money in a savings account so we might end up having squeezed a few extra pounds out of the mess. Still pretty stupid, though.
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Not so much officaldom, but still... in claims for Income Support, it's often important to know who the father of the children is. This has led to some interesting tales, apparently. Not really got a chance to extract many out of Mum yet, but she told me of one claimant who couldn't identify the father because "he still had his bike helmet on at the time"...