Dr. Bancroft was born in Vermont, educated in Concord, N.H., at the Phillips Andover Academy, and received an A.B. (1874) and M.D. (1878) from Harvard. Thereafter, he engaged in general practice in Boston and as a House officer at Boston City Hospital. In 1882, he was appointed superintendent of the New Hampshire State Hospital, succeeding his father, who had been superintendent for 25 years. He remained at the hospital until 1917.
Dr. Bancroft continued a movement started by his father to transfer mental patients from local asylums to state care. In 1888, he established a training school for nurses at his hospital. He was active in professional organizations as a member of the New England Society of Psychiatry and Neurology and the Boston Society of Psychiatry.
For three years, he served as a consultant in mental hygiene at Dartmouth College, served on the State Board of Charities for New Hampshire, and also served on the Board of Trustees for the State Hospital after he left the post of superintendent. Dr. Bancroft was President of the American Medico-Psychological Association (1907–08).