Discovery Channel had a wonderful program this past week about the efforts of a man who works in digital animation to create a 3-D digital image of Christ by using imagery from the Shroud. When the completed image was shown, I teared up. I am pretty sure I saw him do the same at one point.
The scientist went back to the pictures of the samples, and realized that the couple was right, and that the Carbon 14 dating was likely incorrect, because it was picking up the readings off the newer threads. He thought that new testing should be done, but he died of cancer before anything could occur.
There are samples of the Shroud, from an area that was not part of the re-weaving, though, that could be Carbon dated.