Fibonacci numbers are said to pop up all over the place. But most of the claims seem to be inaccurate, unproven or even wishful thinking. Claims of the innate aesthetic beauty of the Golden Ratio usually fall into this category.
But you can still find these numbers in many places. It's not so much a mystical thing as a natural consequence of a particular law of growth. The following seems to hold in real life, and to arise quite naturally from a plausible law of growth.
Take a nice pinecone. It has a base (where it was connected to the tree) and a tip (the opposite end). Look at the cone from its base. The "petals" will form 2 spirals: one proceeding clockwise, the other counterclockwise. Each spiral has several arms. Amazingly enough, if you count the number of arms of each spiral, you will (usually) get two consecutive Fibonacci numbers!
You may find it helpful to use some correction fluid to paint a white spot on one of the petals, to help you count.