In 1950, Bleaney was elected a Fellow of the Royal Society (FRS). In 1956, he was appointed Dr Lee's Professor of Experimental Philosophy at Oxford, succeeding Francis Simon who had died suddenly after only a short time in the post. This unfortunately involved a lot of administrative work, and Bleaney stood down as soon as he felt he had played his part. He was appointed Commander of the British Empire (CBE) in 1965 and became a corresponding member of the French Academy of Sciences in 1974. He won the Royal Society 's Hughes Medal in 1962 and the European Holdwek Medal in 1984.