He said that his team and the laboratories they employed took special care to avoid contamination.
Dinosaurs are not dated with Carbon-14, yet some researchers have claimed that there is still Carbon-14 in the bones. Do these data indicate that a more accurate method needs to be derived?
What solutions are available for increasing accuracy of the tests? From the source linked above: Carbon-14 is considered to be a highly reliable dating technique.
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.
Carbon is naturally in all living organisms and is replenished in the tissues by eating other organisms or by breathing air that contains carbon.