Coal and diamonds both contain radioactive carbon despite supposedly being too old. The half-life of carbon-14 is 5730 years. So after 5700 years there is half the original amount of carbon-14 left. After 11500 years a quarter of the original amount. If we assume all the carbon in an object is carbon-14 (it is however only a fraction) we can get an extreme upper bound on how old it can be (carbon-14 decays into nitrogen so the known presence of carbon-12 lowers this upper bound considerably). If we measure any carbon-14 we know it must be younger than the extreme upper bound age, though it could be much younger.
So carbon-14 has been found in