Buckley's old-fashioned way of speaking wasn't too far from the British-influenced mid-Atlantic accent, which the Hollywood studios taught to actors in the 1930s and '40s. You'll pick up some of the same pronunciations and cadences from recordings of Franklin D. Roosevelt*, as well as Katharine Hepburn—who was, after all, from a wealthy Connecticut family, like Buckley.
This was most interesting because I was thinking about Hepburn when the subject of his accent came up--I always wondered why SHE talked that way. Now I know!