Spurred by the question of the maximum allowable energy for the operation of the Large Hadron Collider (LHC), we have progressed in the understanding of the thermo-electric behavior of the 13 kA superconducting bus bars interconnecting its main magnets. A deep insight of the underlying mechanisms is required to ensure the protection of the accelerator against undesired effects of resistive transitions. This is especially important in case of defective interconnections which can jeopardize the operation of the whole LHC. In this paper we present a numerical model of the interconnections between the main dipole and quadrupole magnets, validated against experimental tests of an interconnection sample with a purposely built-in defect. We consider defective interconnections featuring a lack of bonding among the superconducting cables and the copper stabilizer components, such as those that could be present in the machine. We evaluate the critical defect length limiting the maximum allowable current for powering the magnets. We determine the dependence of the critical defect length on different parameters as the heat transfer towards the cooling helium bath, the quality of manufacturing, the operating conditions and the protection system parameters, and discuss the relevant mechanisms.