Most readers will have heard of the Shroud of Turin. It is a sheet of fabric which had traditionally been believed to have been the burial shroud of Jesus. It has impressed upon its surface a scorched image, the details of which are uncannily congruent with the image made by a man who had been scourged and crucified. The amazing thing about the shroud was that there was believed to be no way that a medieval fabricator could have created such an image.
However, in 1988 the shroud was subjected to a C-14 analysis which dated it to the 13th or 14th century, and that seemed to end the controversy over its authenticity. The radiocarbon tests were regarded by most experts as dispositive. A cloth produced over a thousand years after Jesus' death obviously couldn't have been used to bury him, but lately more questions have been raised.
So much about the shroud indicated a Middle Eastern provenience and a much earlier date of manufacture that some scholars refused to submit to the conclusions of the carbon dating analysis. The image on the shroud was just too difficult to explain in terms of a late medieval forgery and there is now some reason to think that the sample from which the radiocarbon was taken was obtained from a piece of fabric which had been used to patch or repair the shroud about six centuries ago.
Whatever the case, the debate over the shroud's authenticity seems to have been (ahem) resurrected. You can read more about it here.