Depends on the lab, sample, and methods.
There are experiments for the AMS (Accelerator Mass Spectrometry) method going back toward 80,000 years, but that is experimental, not yet a standard, commercially-available practice.
For some materials, contamination is a problem, and I would be careful. Charcoal can absorb humic acids, which, while the labs try to remove them, can be a problem with extremely old samples.
The smaller the quantity of remaining 14C (i.e., the older the sample) the more care needs to be taken with sample selection. That's why creationists are able to find dinosaur bones which date 35,000 or so years old--they get samples bone contaminated by groundwater. They love such contamination! Real scientists do their best to get clean samples. (How many samples did the creationists obtain with results like >50,000 years before they got one contaminated enough that they liked the result?
Lesson: Never rely on only one sample! If you have an old and important specimen, do several or many different samples and use a couple of different labs. Make sure the sample is not contaminated. (If its just a piece of shell from a 3,000 year old site, no big deal. The dates will come out just fine.)
And never, never, ever listen to the young earth types or creation websites when it comes to this kind of science. Their belief blinds them to actual data, and they will stretch things any old which way to try and match their beliefs. Scientists are not like that; if the data heads in an unexpected direction, we can accept it.
As Heinlein noted,
Belief gets in the way of learning.Robert A. Heinlein, Time Enough for Love, 1973
One very persuasive experiment they did was to look at the 'age' as a function of the amount of carbon used. If using more sample results in the sample looking 'older', you have a problem!
(My wife is a high vacuum scientist. When I told her about the sort of thing Humphreys and coworkers were trying to do, she laughed, and said 'don't these guys understand systematic errors?')