I would say that for outdoor sports you'd probably be better off with a 300mm. Yeah, 200mm can be too short for some things, and 400 is probably overkill. 300 is just about right from shooting some of my cousin's soccer games. If you're shooting baseball and wanted to get to an outfielder, you'd most definitely want a 300.
Just be forewarned that both of these lenses are "slow" though, with f/5.6 maximum apertures at the long end. In marginal light or at night you'll have to push the ISO big time (lower quality, grainier photos) to keep the shutter speeds up to keep the players frozen when they're moving. Otherwise you'll have sharp grass but blurry moving players. That's what the f/2.8 (2 stops faster) professional level telephoto zooms are for.
