There was an R/C plane that flew trans-atlantic; it wasn't much larger than a hobby store plane. Cross country is much, much farther (about twice as far as the transatlantic flight), but this is what makes the problem so much fun.
(link: http://news.bbc.co.uk/2/hi/europe/3145577.stm)
(Picture of the plane: http://newsimg.bbc.co.uk/media/images/39393000/jpg/_39393283...)
>Also, your radio won't be able to communicate with the plane for more than a hundred meters legally and a few miles illegally.
The point is to make these things happen autonomously. Beyond that, yes, it is absolutely possible to control the thing all over the world (albeit this would be illegal). While it absolutely wouldn't work for doing real-time controls, HAMs have been bouncing radio waves around the planet for longer than I've been alive.