Good stories rarely begin at the beginning. Though biographies ("He was born in 1837...") and histories ("In the beginning, God created...") may be exceptions, the most engaging narratives draw us into the middle of the action after many events have already come to pass. It's no wonder that George Lucas began his now-famous space opera with Episode IV, and even that first film opened in the middle of a dramatic starship battle.
Game designers often forget this lesson from the storytellers of other disciplines. In an age when the board game market is saturated with "engine building" mechanisms, many suffer from designers who wish to start their game play from the very beginning. While I can see the appeal of building something out of nothing during the course of the game, the result is often a very long, tedious prelude that drains the game of excitement long before things start getting interesting.
However, when a game allows the players to begin with something, it places them in the middle of the story. This can be achieved by something as simple as setting up starting positions at the beginning of a game like The Settlers of Catan. Or it can be as complex as the variable player powers of the different civilizations in games like Cosmic Encounter or Peloponnes. Each of these examples fast-forwards the game's story so that the players can get to the more exciting part of the game in a timely fashion.
I recently learned this lesson when working on my own engine-building game, which is scheduled to be released soon. There are lots of options for building up different areas--both on the board and in a player's own "tableau"--and I initially had the players begin with nothing. After all, that gives players the most control over their destiny, I assumed. After repeated playtests, however, I realized that the first few steps of the game were similar enough for each player--and they were largely unexciting--that I could bypass this "prelude" and immerse them in the middle of the story at the very beginning. All it took was an array of starting resources (and there are many in this game). It sounds simple, but it made a huge impact on the game.
Some designers may be worried that, by jumping ahead in their game, they may have more difficulty maintaining a good story arc so that the tension has room to steadily build to its dramatic climax at game end. It's true, after all, that if you start out at a snail's pace, it's easier to increase the tempo from there. But I found that it is much more exciting to have that tension there from the beginning. That's what makes a good story in other mediums, and it can only improve the story the players are creating within the framework of a board game design.