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Wikipedia Text: Radiocarbon dating
In 1988, the Holy See permitted three research centers to independently perform radiocarbon dating on portions of a swatch taken from a corner of the shroud. All three, Oxford University, the University of Arizona, and the Swiss Federal Institute of Technology agreed with a dating in the 13th to 14th centuries (1260-1390), although recently published chemical analysis (see below) indicates that the sample used was invalid (people think that the material used may have come from one of the patches used to repair it from fire in 1532 - all the patches were removed during a restoration in June 2002). The scientific community had asked the Holy See to authorize more samples, including from the image-bearing part of the shroud, but this request was refused. One possible account for the reluctance is that if the image is genuine, the destruction of parts of it for purposes of dating could be considered sacrilege. Another possible explanation is a reluctance to have the shroud definitively dated.
Radiocarbon dating under typical conditions is a highly accurate science, and for materials up to 2000 years old can produce dating to within one year of the correct age. Nonetheless, there are many possibilities for error as well. It was developed primarily for use on objects recently unearthed or otherwise shielded from human contact until shortly before the test is conducted, unlike the shroud. Dr. Willi Wolfli, director of the Swiss laboratory that tested the shroud, stated, "The C-14 method is not immune to grossly inaccurate dating when non-apparent problems exist in samples from the field. The existence of significant indeterminate errors occurs frequently."
The first part of the parenthetical phrase, "(people think that the material used may have come from one of the patches used to repair it from fire in 1532 - all the patches were removed during a restoration in June 2002)" is simply incorrect. See:
Radiocarbon dating
As with the discussion of possible image formation explanations, Wikipedia mixes theories that are scientifically possible with those that are scientifically preposterous.
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