Thompson was born at York and was privately educated before entering Cambridge University. In 1834 he became a fellow of Trinity, in 1853 Professor of Greek (to which a canonry in Ely Cathedral[?] was then for the first time attached), and in 1866 Master of Trinity College. With the exception of the year 1836, when he acted as headmaster of a newly established school in Leicester, his life was divided between Cambridge and Ely. Thompson died in Cambridge, at the Master's Lodge, twenty years after being appointed Master.
Thompson had succeeded William Whewell as Master and proved a worthy successor; the twenty years of his mastership were years of progress, and he himself took an active part in the abolition of tests (in particular the compulsory religious tests) and the reform of university studies and of the college statutes. In Trinity College An Historical Sketch, G.M. Trevelyan notes;
The efforts of men such as Thompson were fundamental in transforming Trinity College and indeed Cambridge University into truly meritocratic institutions.
As a scholar Thompson devoted his attention almost entirely to Plato; and his Phaedrus (1868) and Gorgias (1871), with especially valuable introductions, remained as the standard English editions of these two dialogues for over forty years.
The quote "We are none of us infallible, not even the youngest of us" is attributed to Thompson and is recorded in Collections and Recollections by G.W.E. Russell[?] (1898) and also in Trinity College An Historical Sketch by G.M. Trevelyan (1943). Thompson uttered these words while Master of Trinity at a College Meeting of the Fellowship on 30 March, 1878. As noted above, this was a time of great reformation within the College, willingly supported by Thompson. G.W. Balfour[?], then a junior Fellow of the College, later politician and Secretary for Ireland[?], proposed a revision of the College Statutes. The quote was, as Trevelyan puts it, "directed in a kindly spirit at the reforming zeal of a group of junior Fellows". Incidentally, the motion was seconded by Coutts Trotter[?], one of the most senior Fellows.