If we had put the money into developing Nuclear Pulse Propulsion, and insisted that propulsive uses of nuclear explosions in space would be permissible, we could have had unmanned probes fly through the closest star systems already.

Apparently the upper theoretical limit for NPP is around 10% the speed of light.

Alpha Centauri is about 4.37 light years away, so travel time would be 43.7 years. If launched at the same time as the Voyager spacecraft (ie., in 1977), a probe like that would have made the fly-by of the system in 2020. We know how to make spacecraft that last that long, as both Voyagers are still kicking and would be more capable if not hobbled by decaying RTG's as a power source. NPP allows you to have a huge number of RTG's for the "cruise" portion, and also a nuclear reactor that can be activated during the exploration phase to provide plenty of power to enable relatively high data rates back to Earth.