Planes with nuclear reactors aboard have been flown by the USSR and by the US, but much like you say, Gunther and EvanCat, the outcomes were that it's extremely dangerous and very, very heavy. Neither aircraft were [i]powered[/i] by their reactors, as both were testbeds to see if it was possible. The answers were pretty much as surmised: yes, sort of, but there's not much point having an aircraft with a huge endurance, as the crew couldn't stay up there forever anyway. Also, the radiation shielding was the biggest problem, as it was difficult to effectively shield the crew and keep the aircraft flyable. 
Unsurprisingly, the idea came about in the 1950's around the time the Cold War was really getting going, and shortly before ICBM's became the next step. The idea was it was supposed to increase the ability to reply to impending attacks, and increase the deterrence factor. There's plenty of info about it on the internet in various places.