I mildly hurt my thumb while snowboarding three weeks ago. It still hurts and pretty swollen even with a brace. Why is it still swollen? Go to the doc?

Need to see ortho. If your thumb is still painful and swollen then need to rule out fracture versus ligamentous injury. Will need to see orthopedic surgeon.
Sprain or break. If you have rested, iced, and braced your thumb it might be broken if it remains swollen. I would recommend getting it xrayed especially so the appropriate treatment can be initiated and prevent future problems caused by malformations or healing inappropriately.