Brebis Bleaney was born at 423 King's Road, Chelsea, London, the second son of Frederick Bleaney, a house painter, and Eva Johanne Petersen, born in Denmark. He attended the Cook's Ground School, Chelsea from where he obtained a scholarship to Westminster City School. In 1933, he obtained an open scholarship in science to St John's College, Oxford, to read physics. He graduated with first-class honours in 1937. Bleaney went on to do research supervised by Francis Simon, obtaining a DPhil degree in 1939.