The research and development programme for the Large Hadron Collider (LHC) magnets developed at CERN includes stability studies which are carried out at Saclay in a collaboration between CERN and CEN Saclay. For Nb-Ti magnets cooled by superfluid helium the most severe thermal barrier comes from the electrical insulation of the cable. The thermal behaviour of He II inside the kapton layers cannot be predicted theoretically. However, its participation in the removal of the heat is necessary for ensuring the stability of the conductor with respect to the expected continuous beam losses. Experiments are carried out to measure the difference in temperature between the insulated conductors and a bath of superfluid helium, when the conductors are heated, and to determine the He II paths through the insulation. The measured thermal resistance includes the resistance of the kapton tape which is wrapped around the cable and the resistance of the cooling channels which are formed by the fibreglass ribbon around the kapton. The total measured resistance is much lower than the solid conduction resistance, in spite of weak heat transport in the cooling channels. Results suggest that there might be small leaks in the kapton covering which are very favourable to heat transfer via the superfluid helium.