For me, the easiest way to know how well a game system works for story-based gaming is to consider classic "groups" like in LOTR, The Avengers, Batman & Robin, Doctor Who + Companions, Robin Hood and the Merry Men:
If there is a large disparity in people's ability to accomplish actions in the game, can you still have fun playing? Game systems I have played and run like DWAITAS, Mouse Guard, FATE, My Life With Master, Everway, Fiasco, Call of Cthulhu are Ok with that. If my CoC investigator is no good at anything, I can still have fun playing him. In other systems I have played and run like D&D (2,3,4), GURPS, Hero, Star Wars Saga, definitely not.
In between are systems like Supernatural, Savage Worlds, Deadlands where the answer is more mixed.
One you've played a few rally different systems, it's pretty clear that different systems have different things they are good for. D&D provides little support for story-based games directly, so if you want to run a story-based game with D&D, to me *it makes perfect sense* to look for advice on how to run a game from other systems, and see how much you can use. The stated use of Beliefs, Traits and Instincts by a previous poster is an excellent example.
For me, I used Everyway's card-based resolution as a way to introduce story into high-epic 4e play; each person could choose pictures from anywhere they liked that they felt described their character. If they used the car in conjunction with a standard 4e power, the power was enhanced based on the degree to which the card's picture matched the story situation. Examples:
* One character used a card showing courtly love to enhance his power to move fast; his in-game wife also got the benefit, dispelled her immobility and they escaped.
* One character had a card showing a door being closed by a wizard. The plot I had planned was to force them to use a powerful artifact to close a gate to the Far Realms. He used the card with a simple "I close the door" action and that changed the campaign.
* One character had a picture of Sigil. The characters had a near-finale battle is Sigil, and the Lady of Pain ended up dead. The character who owned that card was asked to take over running Sigil, and did so,
Adding story elements to a game that doesn't directly support them isn't hard -- all it takes is an open mind, a willingness to read other systems and get ideas, and a set of players who can handle a GM telling them: "I'm going to try this. If it doesn't work, we'll drop it, but let's give it a go ..."