There are some really great lofty mathematic principles in the Fibonacci sequence. But let's talk real world. The Fibonacci sequence appears all over the place in biology. To start out, let's look at the original problem again. It was based on some rabbits under ideal conditions with a lot of inbreeding. It's not terribly realistic.

     Oddly enough, though, the sequence shows up in other populations. If you look at a honey bee's family tree in reverse you can find it.

     First, a few fun facts about honey bees:
          Every hive has:

One female, which is called the Queen.
Many female workers, which produce no eggs.
Some males, which are drones. They result from unfertilized eggs.

     Alright, that was educational. So, now let's apply that with. If we take a male bee, then he has one parent, because he resulted from an unfertilized egg. That one parent that laid the egg must have been a female, so she had two parents. Therefor, the male bee has two grandparents. Of those grandparents, one was a female and had two parents, and one was a male and had one parent. So our original bee had three great grandparents. This just goes on and on until we're all thoroughly confused. The same holds true in the case of a female bee.