I'd always assumed that if a cow died of any cause other than proper shechita (kosher slaughter), the meat is neveila. If it died by kosher slaughter but had already been seriously injured or diseased, it's treifa.
There's an "old manuscript Rashi" printed in some editions in the margins of Zevachim 70a that speaks of "a treifa that became a neveila", for which one could be punished for both.
Has anyone ever heard of this? Do other rishonim agree? It was news to me!
It looks like this is in fact the subject of a machlokes between R' Yochanan and Reish Lakish (Yerushalmi Nedarim 6:1 (26a)). R' Yochanan says that one who eats "a treifah that became neveilah" is indeed punishable for both prohibitions. (Although Korban Ha'eidah actually reverses the two opinions and attributes this view to Reish Lakish, since he says it depends on what verses these two prohibitions are derived from.)
Pnei Moshe there spells out that indeed according to this view, the fact that it became neveilah doesn't take away the animal's designation as treifah.
Imrei Baruch (at the foot of the page there) adds that this seems to depend on the question of whether the prohibition of treifah applies while the animal is still alive, and references Tosafos to Chullin 32a ד"ה ורמינהו (where indeed they say the same as Pnei Moshe) and a related sugya in Chullin 103a, where indeed Reish Lakish says that this prohibition applies only after the animal is slaughtered.
