I was "overpaid" by many thousands of pounds. At the time that the money was paid they were adamant that it was correct even though I queried it numerous times. As it happens, I think that it was NOT an overpayment, but that was sort of irrelevant. When they then decided that they had overpaid me and wanted the money back I told them that I could not afford to do so and since they had made the mistake, it was their problem. I had moved to the usa by this time, so there weren't any more payments for them to take the money out of luckily, but it did involve many many many conversations and 3 letters and a form (filled in twice, I think). The whole process took a year and ended when I finally sent them a scathing letter with all the dates of the times I had called them to query the original "over"payments, all the times I had called to contest the repayment request, copies of all the letters I had sent and I think I even included a threat to commence legal action for the harassment (very american, I know). Anyway, they did finally conceed that the "mistake" was theirs and waived the repayment requirement.
My understanding is that if you can show that the mistake was theirs and not your, then they will not make you repay the money. However, it is very hard to prove that the mistake is theirs. My advice is that it's probably not worth the hassle unless the amount is significant.