In short, no. However in many games you can get one of the starter pokemon and migrate/trade for others.
- choose one as starter in Diamond/Pearl/Platinum.
- starter in Ruby/Sapphire/Emerald then migrate
- get one from Steven (at Pokemon Fan Club in Vermillion) in HeartGold/SoulSilver after beating the Elite Four and Red
- choose one in Emerald after completing the Pokedex then migrate
- starter in HeartGold/SoulSilver
- starter in FireRed/LeafGreen then migrate
- get one from Professor Oak in HeartGold/SoulSilver after defeating the Elite Four and Red