C-14 has a half-life of about 5,730 years, and like most radioactive decay and half-life based dating methods, it's only good back about seven half-lives. So radiocarbon dating can only generally be used for organic-based things back to about 35,000 - 40,000 years old, i.e. about seven half-lives. It can be extended back a little further to maybe 50K-60K years, too, but with less accuracy.
Regarding the accuracy of radiocarbon dating, depending on the quality of sample, you might have +/- accuracy of just a decade or two for a really good sample. But it also depends on the radiocarbon calibration curve; i.e. for a sample of a given quality, it may have different +/- accuracy results because of wiggles in the radiocarbon calibration curve. With poor samples and contamination, you could be looking at +/- accuracy of hundreds or even thousands of years. With multiple good samples, wiggle matching of the radiocarbon calibration curve, Bayesian techniques, etc., one could get down to +/- accuracy of just a few calendar years.
Back to the press release at hand--the plant/organic material they attempted to age date with radiocarbon was radiocarbon dead, i.e. no more measurable radiocarbon remained. They used other dating methods, too, including cosmogenic Be-10 dating (a speciality of Bierman, one of the authors) and luminescence dating as well, but they just very lightly mentioned these in the press release (not using the specific terms I have used above).