Answer
The three laboratories that carbon dated the Shroud of Turin could only give a range from 1250 to 1390 CE for the production of the Shroud of Turin.
However it is known that the Shroud already existed by 1355, when it first appeared at a little church in Liry, in north-central France. Its owner, a soldier of fortune named Geoffroy de Charney, claimed it as the authentic shroud of Christ, although he was never to explain how he acquired such a fabulous possession. That knowledge improves the carbon dating results to the range 1250 to 1355 CE. A Bishop's report, written in 1389 by Pierre D'Arcis to the Avignon pope, Clement VII, spoke of a predecessor who conducted an investigation and uncovered the forger: "Eventually, after diligent inquiry and examination, he discovered the fraud and how the said cloth had been cunningly painted, the truth being attested by the artist who had painted it, to wit that it was a work of human skill and not miraculously wrought or bestowed."
So we know from D'Arcis that the Shroud was certainly produced within the range of dates given by carbon dating, but not with great accuracy.