I know Apple doesn't officially support installing operating systems that predate the one that ships with the hardware, but is it possible (perhaps through adding drivers?) to run Snow Leopard on a mid-2011 Macbook Air?
Currently no. There are known issues with the graphics drivers and glitches in the software as a result. You can get SL to install, but the performance and stability are lacklustre at best. And Apple is unlikely to add driver support for SL in future revisions of that operating system.
Additionally, you cannot manually add drivers for core components to OS X. You may be able to modify kernel extensions, but not at the level required to get the old OS operational. You may be able to port the graphics driver from Lion, but that is certainly a long-shot.
You can follow a discussion over the the Apple Forums for additional information, where it has been reported that the new Air installs a special build of Lion that carries with it a unique graphics driver. One user claims they got SL installed, but it does not recognize the internal display. That's a pretty big show-stopper, unless you want your new Air to be a desktop.