I think it has to do with the phrase, “the third time’s the charm.” If you think of other fairy tales, the number three is pretty significant. For example, Cinderella goes to the ball three times but the third night is the night she decides to stay late, which results in her leaving her shoe and him finding and marrying her. In The Three Little Pigs, it’s the third pig that figures out how to get rid of the wolf, and in Goldilocks and the Three Bears, it’s the baby bear’s things that she tries out third each time, and those are what she ends up finding just right.
In Big Fish, everything happens after the third year: Edward finishes growing and is able to leave his bed, he successfully learns Sandra’s name after his third year of working in the circus, and he gets sick after three years of not talking to his son, which prompts Will to return and try to reconcile with his father.
Maybe the use of threes in Big Fish is utilized to connect the movie to fairy tales, especially because each of Edward’s stories seem like fairy tales themselves and how significant the number three is in fairy tales. I looked up a list of fairy tales on Wikipedia (http://en.wikipedia.org/wiki/List_of_fairy_tales) and of those titles, 27 of them had the number three in the title. I’m still not sure of what the significance of three is exactly, but I think it’s used to further establish the connection between Big Fish and fairy tales.