Bury was born and raised on 16 October 1861 in Clontibret, County Monaghan, as the son of Edward John Bury, where his father was Rector of the Anglican Church of Ireland and Anna Rogers. He was educated first by his parents and then at Foyle College in Derry. He studied classics at Trinity College Dublin, where he was elected a scholar in 1879, and graduated in 1882. He was elected a fellow of Trinity College Dublin in 1885 at the age of 24. In 1893, he was appointed to the Erasmus Smith's Chair of Modern History at Trinity College, which he held for nine years. In 1898 he was appointed Regius Professor of Greek, also at Trinity, a post he held simultaneously with his history professorship. In 1902 he became Regius Professor of Modern History at the University of Cambridge.