Carbon dating of dinosaur fossils
The theoretical limit for C-14 dating is 100,000 years using AMS, but for practical purposes it is 45,000 to 55,000 years.If dinosaur bones are 65 million years old, there should not be one atom of C-14 left in them.In particular, it is implausible that it would have been considered worthwhile to try to use radiocarbon dating methods on these bones, since the rocks that they were taken from were determined to be 99 million years old, as shown in this paper by Kowallis et al.Now, it is known that $^\text$ decays at a fast enough rate (half-life ~6000 years) for this dating method to be absolutely useless on such samples. would not have been able to obtain this sample, had they been honest about their intent.
"Comparing such different molecules as minerals and organics from the same bone region, we obtained concordant C-14 results which were well below the upper limits of C-14 dating.
The CRSQ study authors tested seven dinosaur bones, including a from Montana, hadrosaurids, a cartilaginous paddlefish, a bony fish, and fresh-looking wood and lizard bones from Permian layers in Canada and Oklahoma.
Five different commercial and academic laboratories detected carbon-14 in all the samples, whether from Cenozoic, Mesozoic, or Paleozoic source rocks. The team also compared the results to several dozen published carbon-14 results for fossils, wood, and coal from all over the world and throughout the geologic column.
The age that these groups claim to find is usually on the order of thousands or tens of thousands of years old.
The particular example you bring up is one of the most famous such cases.
At a horizon of 40,000 years the amount of carbon 14 in a bone or a piece of charcoal can be truly minute: such a specimen may contain only a few thousand 14C atoms.