The procedures used to isolate and analyze the parent and daughter nuclides must be precise and accurate.
This normally involves isotope ratio mass spectrometry.
After an organism has been dead for 60,000 years, so little carbon-14 is left that accurate dating can not be established.
On the other hand, the concentration of carbon-14 falls off so steeply that the age of relatively young remains can be determined precisely to within a few decades.
For most radioactive nuclides, the half-life depends solely on nuclear properties and is essentially a constant.