Okay, I am very confused right now. I am fairly certain that, a while back in the game, I answered all the Dragon Master guy's questions right and earned the ExtremeSpeed Dratini, but had a full party, and decided to come back later. I dunno, maybe I made that memory up. Either way, now all he does is say something irrelevant about Lugia. I even went out and caught Lugia, put it in my party, and that's still all he does when you talk to him now. No mention of the Dratini. And I've double- and triple-checked, and that Dratini is nowhere to be found in my PC.

I really want that Dratini! Does anyone know what's going on? Is it gone forever now? And if so, does anyone have one they're willing to trade for something?

I had this problem too. You need a free space in your party, and you need to go back into Blackthorn Gym, and talk to Clair. Go back to the dragon master and he should give you Dratini.