He was the son of the Rev. John Bury (1580–1667), and matriculated at Exeter College, Oxford, on 5 April 1639, aged 15. He took his degree of B.A. on 29 November 1642, was elected a Petreian fellow of his college on 30 June 1643, and become full fellow on 6 May 1645. A royalist, he refused to submit to the parliamentary visitors of the university, and took refuge with his father in Devon. At the Restoration of 1660 he was restored to his fellowship. In 1666 the rectorship at Exeter College became vacant, and Bury was elected (27 May), on the recommendation of Archbishop Gilbert Sheldon and with heavy-handed support from Charles II. On 22 June in the same year he took the degree of B.D. and five days later became D.D.
