Protons and carbon ion beams for hadron therapy can be delivered by cyclotrons with a fixed energy. In order to treat patients, an energy degrader along the beam line will be used to match the particle range with the target depth. Fragmentation reactions of carbon ions inside the degrader material could introduce a small amount of unwanted contaminants to the beam, giving additional dose to the patient out of the target volume. A simulation study using the FLUKA Monte Carlo code has been carried out by considering three different materials as the degrader. Two situations have been studied: a realistic one, lowering the carbon beam energy from 300 MeV/n to 220 MeV/n, corresponding to a range of 10 cm in water, and the worst possible case, lowering the carbon energy to 50 MeV/n, corresponding to the millimeter range. The main component of the contaminant is represented by alpha particles and protons, with a typical momentum after the degrader greater than that of the primary beam, and can be eliminated by the action of a momentum analyzing system and slits, and by a second thin absorber. The residual component of fragments reaching the patient is negligible with respect to the fragment quantity generated by the primary beam inside the patient before arriving at the end of the target volume.