How can you flawlessly confirm how old something is?

If something was etched into the wall of a cave do you really trust the number scientists put on it? I once heard about a study where they divided a fossil into two separate parts and sent them off to two different labs. One lab confirmed that the rear of the fossil was 10,000 years old while the other said the head was 20,000 years old. So which test was right, or are they both wrong?