When she was selected for the 1964 British Olympic team Packer worked as a physical education teacher at Coombe County Girls' School, New Malden, Surrey. At the Olympics she shared a room with long jump gold medallist Mary Rand. Packer was hoping to win the 400 metres, but was beaten into second place by Betty Cuthbert of Australia, despite setting a new European record at 52.20 seconds. Disappointed, Packer planned to skip the 800m event and had a shopping trip instead, until her fiancé, Robbie Brightwell persuaded her to compete. Before the Olympics, Packer only had five domestic 800m races; she had taken up a longer distance to improve her stamina, and earned the third British spot at the last minute.