The preferred method of dating dinosaur fossils is with the radiometric dating method. And the result of this accepted method dates dinosaur fossils to around 68 million years old.

However:
Consider the C-14 decay rate. Its half-life ($t_{1/2}$) is only 5,730 years—that is, every 5,730 years, half of it decays away. The theoretical limit for C-14 dating is 100,000 years using AMS, but for practical purposes it is 45,000 to 55,000 years. If dinosaur bones are 65 million years old, there should not be one atom of C-14 left in them.

Dinosaurs are not dated with Carbon-14, yet some researchers have claimed that there is still Carbon-14 in the bones.

So what needs to be done about this inconsistency? Do these data indicate that a more accurate method needs to be derived? What solutions are available for increasing accuracy of the tests? Or do we need another dating method all together?

Considering Contamination

From the source linked above:


  Carbon-14 is considered to be a highly reliable dating technique.  It's accuracy has been verified by using C-14 to date artifacts whose age is known historically. The fluctuation of the amount of C-14 in the atmosphere over time adds a small uncertainty, but contamination by "modern carbon" such as decayed organic matter from soils poses a greater possibility for error.
  
  Dr. Thomas Seiler, a physicist from Germany, gave the presentation in Singapore.  He said that his team and the laboratories they employed took special care to avoid contamination. That included protecting the samples, avoiding cracked areas in the bones, and meticulous pre-cleaning of the samples with chemicals to remove possible contaminants.  Knowing that small concentrations of collagen can attract contamination, they compared precision Accelerator Mass Spectrometry (AMS) tests of collagen and bioapatite (hard carbonate bone mineral) with conventional counting methods of large bone fragments from the same dinosaurs. "Comparing such different molecules as minerals and organics from the same bone region, we obtained concordant C-14 results which were well below the upper limits of C-14 dating. These, together with many other remarkable concordances between samples from different fossils, geographic regions and stratigraphic positions make random contamination as origin of the C-14 unlikely".



Question: Am I supposed to have no idea what's going on in the first few hours of play?
Answer: Bioshock: Infinite starts in medias res.  There are a lot of details of the "hows" and "whys" that will be explained as you play.

From the outset, you should know that the player character is Booker DeWitt, a private detective who has been hired by a pair of rather unusual people to retrieve a rather unusual girl from a rather unusual city.  

The game is quite linear, so there's little to no chance that you'll get lost and not be able to figure out what to do next.  Just keep following along, and eventually the plot will start to come into focus.

However, the plot can be relatively difficult to understand, so I won't go so far as to say that eventually things will make sense :)

There is little to no dependency between Bioshock 1/2 and Infinite.  If you haven't played the first two, you might miss out on some understanding of a few late game moments, but it's nothing major.  I believe the "Burial at Sea" DLC is more tightly tied to Bioshock 1, so you might want to play that game first if you plan on playing the DLC.