yes, but it's usually exclusive distribution rights, which are pretty major. you also usually have to alter your game to put someone's logo / link to their site in it. i also haven't really heard of a game that was sponsored *and* sold to customers, can you think of any examples? usually they only sponsor free games.
You only sell Web distribution rights. Yes, the game you get sponsored must be free, but you can also have a version that is commercial provided it's not a distributable Web game.
Examples: Steambirds, Doodle God, Karoshi Factory, Warlords and Solipskier.
All of these sell on iPhone, but I'm pretty sure you can sell a PC downloadable too, though, obviously, you'd have to include "added value".