Archaeology and radiocarbon dating
First, any instrument which is built to measure radiocarbon has a limit beyond which it cannot separate the signal due to radiocarbon in the sample from the signal due to background processes within the measuring apparatus.
Even a hypothetical sample containing absolutely no radiocarbon will register counts in a radiocarbon counter because of background signals within the counter.
It is not difficult to see how such a claim could arise, however.
There are two characteristics of the instrumental measurement of radiocarbon which, if the lay observer is unaware, could easily lead to such an idea.
MYTH #2 Radiocarbon dating has established the date of some organic materials (e.g., some peat deposits) to be well in excess of 50,000 years, thus rendering a recent creation (6 to 10 thousand years ago) impossible.
Some organic materials do give radiocarbon ages in excess of 50,000 "radiocarbon years." However, it is important to distinguish between "radiocarbon years" and calendar years.
For this reason special precautions need to be exercised when sampling materials which contain only small amounts of radiocarbon.
Radiocarbon is not suitable for this purpose because it is only applicable: a) on a time scale of thousands of years and b) to remains of once-living organisms (with minor exceptions, from which rocks are excluded).
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.
The shells of live freshwater clams can, and often do, give anomalous radiocarbon results.
However, the reason for this is understood and the problem is restricted to only a few special cases, of which freshwater clams are the best-known example.