Long tree-ring chronologies are rare (there are only two that I am aware of which are of sufficient length to be of interest to radiocarbon) and difficult to construct.
They have been slowly built up by matching ring patterns between trees of different ages, both living and dead, from a given locality.
Other radiometric dating methods such as potassium-argon or rubidium-strontium are used for such purposes by those who believe that the earth is billions of years old.
In the early days of radiocarbon analysis this limit was often around 20,000 radiocarbon years.
Thus, all the researcher was able to say about samples with low levels of radiocarbon was that their age was greater than or equal to 20,000 radiocarbon years (or whatever the sensitivity limit of his apparatus was).
It is not correct to state or imply from this evidence that the radiocarbon dating technique is thus shown to be generally invalid.
The problem with freshwater clams arises because these organisms derive the carbon atoms which they use to build their shells from the water in their environment.
If this water is in contact with significant quantities of limestone, it will contain many carbon atoms from dissolved limestone.
Since limestone contains very little, if any, radiocarbon, clam shells will contain less radiocarbon than would have been the case if they had gotten their carbon atoms from the air.
This gives the clam shell an artificially old radiocarbon age.
This problem, known as the "," is not of very great practical importance for radiocarbon dating since most of the artifacts which are useful for radiocarbon dating purposes and are of interest to archaeology derive from terrestrial organisms which ultimately obtain their carbon atoms from air, not the water. Samples of coal have been found with radiocarbon ages of only 20,000 radiocarbon years or less, thus proving the recent origin of fossil fuels, probably in the Flood.
It is not difficult to see how such a claim could arise, however.