In 1949, Brebis Bleaney married Betty Isabelle Plumpton at St Nicolas Church, Guildford. Betty Plumpton had been his student; they published a book together: Electricity and Magnetism, in two volumes (Oxford: Clarendon Press, 1957). The couple had two children, Michael Bleaney, now a noted economist, and Carol Heather Bleaney, who was at SOAS University of London and has published widely on the Middle East.