We have measured total nuclear disintegration cross sections for 208 Pb ions at 33 TeV ͑160-GeV A͒ colliding with C, Si, Cu, Sn, and Pb. Using well established theory, we calculate the nuclear electromagnetic, electron electromagnetic, and the hadronic contributions and find that their sum underestimates the measured cross sections. An additive correction term linear in target Z T (i.e., 120Z T mb) is necessary to bring agreement between theory and experiment. The source of this additional term is unknown. [S0031-9007(97)04384-6]