Kibble was born in Madras, India, and was the grandson of William Bannerman, an officer in the Indian Medical Service, and the author Helen Bannerman. He was educated at Doveton Corrie School in Madras and then in Edinburgh, Scotland, at Melville College and at the University of Edinburgh. He graduated from the University of Edinburgh with a BSc in 1955, MA in 1956 and a PhD in 1958. Kibble was married to Anne Allan from 1957 until her death in 2005. They had three children.