Even a small amount of c14 from a contaminant can produce an incorrect date in an old sample. The ages derived are compared with another, and usually, there has been good agreement between the methods. Again, this is really just beyond the c14 limit for sites such as these. This is a difficult one, because we can date pretty much anything from today or in modern times, but getting an actual 'date' is hard. In the 1950s and 60s, people blew up alot of nuclear bombs, and one thing that happened because of this was that alot of radiocarbon was created in the air artificially.
Radiocarbon is a side effect of nuclear bombs.
It sounds bad, and nuclear bombs are not pleasant when they are generated, but for science there have been some spinoffs because we have been able to study the movement of this radiocarbon through the environment and learn alot about how radiocarbon is transported naturally. So this has been beneficial. We can also date things that have happened since 1950 rather well because of the sudden jump in radiocarbon on Earth, so that it is possible to figure out within 2-3 years sometimes, the date of a sample. Generally, we can date things pretty well over the past 1000 years, it becomes difficult from about 1700 AD to 1900 AD because of natural changes in radiocarbon, and since 1950 AD dating is quite possible. The Dead Sea Scrolls are a very famous archaeological discovery, and many have been dated by using AMS at the Arizona AMS laboratory.