Thermonuclear pulse propulsion (a la Project Orion) can do better than that- up to 0.1c. And that requires no unproven technology; just a lot of money and a lot of politics (and a lot of fission fuel to buy with that money; I dunno if it's more than you could actually get on the market or not, but it would definitely cost A Lot).
IIRC you're talking on the order of several hundred nuclear bombs for a single interstellar trip, i.e. a significant fraction of the current US warhead supply, but by no means the whole lot.
I think intersteller travel at sensible speeds would require rather more energy than that - the relevant Wikipedia page gives estimates (depending on the approach used) of 30 million to 300 thousand 1MT bombs which is, fortunately, a few orders of magnitude larger than any nuclear arsenal:
