Alaska Brian has made an important point, I think. The Green Party (especially in Germany) acquired political power in a similar way. It took some time and commitment maybe more then most Americans have—we tend to be very myopic—but they succeeded. I’m not sure if this can happen in the US, but the “grassroots” is probably the place to start.

Third parties have found significant support in America too, but they were driven by personality (a bad long term strategy), and that made them less sustainable (e.g. Ross Perot and the Reform Party, Teddy Roosevelt and the Bull Moose party, maybe there were others).