I think I'll be satisfied with a year, although staying longer would definitely benefit my Japanese skills more. I wouldn't want to leave everything else behind for any longer than that though.
I think you would have to either be completely disinterested or in a very urban area where you can always find English speakers to hang with to not pick up anything. But yeah, more important situations like talking to a doctor would be pretty hard and frustrating...
I went to all the tourist areas and big cities (Tokyo, Osaka, Kyoto, Nara, Hiroshima, Hakone), so the level of English was probably a lot higher than in other places. People were very patient with my disjointed Japanese words though (my grammar is a bit terrible) and didn't have much trouble. Then again they're probably a lot more used to clueless tourists than the more rural areas that JET will send you to.
My theory is that if other people with zero Japanese skills can do it, then I sure can
I'm under the impression that you have a supervisor or someone to ask questions from if you get really stuck, right?