Fame
It's certainly possible. There are plenty of awards given out each year for costume design on stage, television, and in film, so there's no reason you couldn't have a moment in the sun. Okay, you'd only be famous only to the other 299 costume designers in Hollywood who might care, but it's better than nothing. And that Oscar's sure going to look nice on your mantle.
However, while you might score some podium time when accepting your award, we really can't tell you the names of any award-winning costume designers in the last twenty years without looking them up, so you probably won't be achieving any lasting recognition by anyone other than movie industry insiders. But then, we haven't heard your acceptance speech. It might just knock our socks off.