In recent years, an anomalous increase of faults in underground medium voltage (MV) cable lines has been recorded in Italy, especially during summer; the largest number of faults affected cable joints. The assessment of joint thermal stress, both in normal operation and during faults, is paramount. The study presented in this paper focuses on cable heating effects due to short circuit currents flowing through cable screens during ground faults (e.g., in case of cross country faults, CCFs, whose current values are comparable to line-to-line short circuit), considering the contact resistance (CR) between cable screens and copper stocking due to inaccurate joint manufacturing. A thermal model, already developed and discussed by the authors in previous papers, has been extended and applied in this study in order to assess the CR effects in cable and joint heating during failures. Parametric studies have been carried out on a typical cable-joint system, varying fault current and CR values, as well as considering protection schemes normally adopted by distribution system operators (DSOs) in Italian MV distribution grids. Results show that for CR values larger than few milliohms, fault currents due to CCFs are able to overheat the joint well beyond the maximum tolerable temperature of insulation, thus leading to cable failures when the shortest fault clearing times (i.e., 120 ms) are considered.