The rule of three is one of the most interesting things found in fiction. Think of all the stories you read as a children. The hero or heroine being granted three wishes. The Three Little Pigs and Goldilocks all follow this pattern in their stories. Using the rule of three can beef up a story and make the reader anticipate.
Think of how bland these stories would be without the rule of three. Would the wolf win if he encountered the smartest pig early in the story. And what about Goldilocks. Three dishes of porridge, the first two not right but the third just right. So what does this mean?
One is only an incident. Two forms a pattern. Three breaks it. I've often used this in stories having the hero, heroine or villain attempt something three times. The third will bring defeat or winning depending on what I'm trying to achieve.
This is using the echo factor and expanding it to show the pattern broken. The first time a disaster or a success occurs, the reader will note this and perhaps not pay attention. The second time will make them sit up and take notice. The poor hero or heroine may seem destined to fail and the villain to succeed, Then comes the third with a reverse and all falls into a pattern we've recognized since childhood.
So think about using this technique in a story and see what happens.