A large cryopump has been operational since 1994 inside the plasma chamber of JET (Joint European Torus), the world's largest thermonuclear fusion facility. In this work, an analytical model, which analyses the uncontrolled warm-up behaviour of the cryopump and associated water cooled components, under all possible operating conditions, is presented. The system behaviour is simulated for a wide range of fault conditions, e.g. loss of vacuum, loss of cryogen flow, loss of cooling water supply and combinations of these. The model is validated by a series of experiments conducted both in a specially designed test rig and inside the JET vacuum vessel, using real scale components. Based on this analysis, an automatic safety system has been constructed and implemented into the JET operation routines. The system incorporates protective actions (e.g. draining) which prevent the development of excessive stresses in the cryopump components, thus ensuring their mechanical integrity and reliability. The work has been divided into two parts, Part I outlines the model used in the analysis and examines the behaviour of the divertor cryopump; Part II deals with a different type of cryopump which operates outside the plasma chamber.