Recently one of the players in a game I run was worried about the character he wanted to bring in. See, we're about to start the 4th generation of my L5R game, and he was worried that his character was a little too "main character" or "protagonist." Essentially, he didn't want to play a character that would take over the game and make the whole thing be all about him.
My response to this was simple: I want every player in my game to make a protagonist. I want a six protagonist pile up. Why? Because if everyone does that. If everyone brings a character that could have the whole thing be about them, then the game gets to be very special as we see all these plot lines collide and see what pieces hit the ground where.
Now, I can do this with my game because I trust my players. We know each other. We've gamed with each other for a long time. All of us are ok if things take a little longer to get going on a personal plot if it means everyone is getting time. We can make characters that can do this.
However, my game isn't special. This isn't something unique to my game. Any game can do this, but as a group your players have to be comfortable with each other. How comfortable? Comfortable enough to push for room in the spotlight with everyone else. Comfortable enough to not be talked down and fight for time when a GM is over worked with everything that is going on.
Encourage your players to do that. Encourage them to come out into the limelight. Give them the time. As the GM, don't be afraid to slow everyone down so you can focus on everyone as you need to.
Give it a shot. You might just be happy with what happens.