Yes, it is very possible. If you're looking to do it without trading though you'd probably need both FireRed and LeafGreen, Emerald, Platinum, HeartGold and SoulSilver, and Black and White. That's because of version exclusives, but this forum has a huge trading area that would be more than willing to trade you pretty much any breedable Pokemon with no hassle because it's not a big deal, so that would cut out one in each of the pairs I listed. In addition, you can trade for the rest if you didn't want to buy all the games, but it might be a bit more difficult to trade for legendaries and you might have to offer something of value in return.
Good luck!
Not all men...
Are all men stupid?
That's right.