Holy shit. 
I am literally in SHOCK reading this thread. Almost every single post is from someone who has been 'overpaid' by tax credits, and has had a demand to pay it all back, and ended up struggling financially! As some people have said, how the FUCK are they allowed to get away with this? Basically I guess - as someone said earlier - they send their nasty-ass fucking poison shit threatening you, and because it's 'the government' you cannot do anything about it really, and are scared into paying it. As has been said, some people end up paying back all that they have been given, and in some cases, even more!!! 
Me and DH claimed when he lost his well paid job some 10 years ago and had to take a lower paid one. We didn't get masses, maybe £40 a week I think. We did the usual, gave the income from the past year, they gave the 'award' based on that amount - even though it's OBVIOUS the next year will be more with pay rises, overtime etc - and of course we ended up with an overpayment. Not as bad as some, but still an overpayment. £700. Although they didn't inform us of it for a couple of years!
Then they reduced our award which left it at £15 a week or something, and then a couple of years later - even though we had only been getting £15 a week, we ended up with another overpayment of £679! 
So then they reduced our 'award' to £6 a week! Then a couple of years after THAT, we stopped being entitled to tax credits at all as our income was too high and we had no kids under 18. During that 2 years, we paid about £600 off the debt (they took it out of our 'award.')
We never heard anything for a year, and then a letter dropped on our doormat saying we owe £899! 
WTAF?
I rang them and queried it, saying 'how can we owe this, you have been taking money off our award for about 4 or 5 years!' They said we owed it from 2009 and 2011 and 2013, and this was all the bits we owed added together as a lump sum!!!
Like many here, I never understood what the fuck had been going on (do THEY even understand?!)
and I couldn't argue the point because I didn't know what to say or how to 'prove' I didn't owe anything!!!
At that point, I agreed to set up a payment plan of £30 a month, and made them write to me with the agreement and confirmation that we owe NOTHING ELSE, other than the £899 (which they did surprisingly!) Then I told them me and DH were not receiving tax credits at the moment, and we never wanted to claim them anymore, and we wished to close down any open claim we had, so we would not be issued them automatically. We said it has caused us nothing but problems, and we ended up worse off financially. So they sent us a letter saying they were closing our claim.
Since we made the agreement to pay £30 a month last summer, we have had THREE FUCKING LETTERS from a debt collection agency, demanding payment in full for not sticking to our agreement, and threatening court action, and that our credit record will be affected!!! (we have been sticking to it!
) When I ring HMRC, they say 'oops, that was sent in error. Ignore it!' WTF!
Cunts.
They have caused so much stress for so many people that it's sickening. I feel for people who owe multiple 1000's.
I have a similar tale to one further back... I know one lady who kept getting £400 a month (instead of the £235 she should have been getting,) because she apparently told tax credits that her husband is disabled (she never did.) She rang and wrote and everything, saying 'he is not disabled, is this right etc etc etc...?' And they just carried on paying.
2 years later, they demanded proof of her husband's disability (which she didn't have, as he wasn't disabled.)
She ended up owing them £4,000! THEY were at fault, yet her family was crippled financially for several years, as they 'adjusted the award' and left her with £80 a month.
Absolutely disgraceful. If a normal business were to operate like this, they would have been closed down!