This story is bogus. Those who submitted the sample were specifically warned that they were dating the organic material on the pottery and NOT the pottery itself.
"We dated the organic material on the stone but warned the
submitter that the date had no meaning until this material had been identified. This was
never done, and yet they still publicized the date which of course provides no information
of the actual date at which the stone was painted."
- Professor Michael Tite, Director of the Oxford Laboratory
This reminds me of the time Ken Ham sent in a dinosaur fossil to be carbon dated. He then went around telling everyone how it came back as only being a few thousand years old. Of course, the one part of the story he didn't reveal was that the company that dated the fossil told him that fossils contained no carbon and that what they dated was the varnish that had been put on it later to protect it.
Anytime you hear someone make quotes about carbon dating fossils, diamonds, pottery, metal, etc. it should set off your BS meter. Carbon dating is ONLY used to date organic material. For other materials there are other dating methods. Furthermore, scientists generally use two different dating methods or more when identifying the ages of things. Because when you have two or more dating methods converging on the same date you can have an extremely high degree of confidence that date is correct.
Carbon dating is a specific tool for a specific kind of material. Carbon dating a fossil or a piece of pottery and then concluding "c14 dating is inaccurate" would be like trying to use a hammer to mow your lawn and then concluding "hammers aren't very good tools."