The specific heat of three samples of YBa2Cu3O7 (123), YBa2Cu4O8 (124) and YBa2Cu3.5O7.5 (247), selected using the criterion of the lowest extrinsic upturn in C/T at low temperature in zero field, was measured from 1.0 to 30 K in various magnetic fields ranging from 0 to 14 Tesla. The analysis of the data shows that the results cannot be described by a magnetic Schottky term, or by any combination of Schottky terms, including a continuous distribution. Therefore, the random Heisenberg antiferromagnetic exchange model of Bulaevskii does not apply. Alternatively, the observed behaviour may be described satisfactorily if one includes a magnetic Schottky term with an amplitude that is proportional to the applied magnetic field. This suggests a localization of quasiparticles in vortex cores. A picture of this kind is consistent with recent measurements of the local density of states in materials with an extremely small coherence length.