Firaxis posted here on the boards saying that Civ3 with some changes to the .ini or .txt files or whatever it is they're using in Civ3 is capable of having all 16 civs in one game. Unfortunately however the same Firaxis poster said that the diplo, trade and other similar screens were not set up for more then 8 civs, 7 AI + 1 you.
So it looks like we'll have to wait for an expansion, patch or mod to fully exploit all 16 civs in one game.