Superconducting magnets can get quite high fields, higher than the limit given by the core electromagnets, as the LHC experiments have demonstrated, but are subject to limitations of the cryogenics and the materials:
Superconducting magnets must operate below both the critical temperature and the critical field of the material from which they are constructed.
Careful design is used to find a fine balance between wire composition, diameter and distribution along the axis of the coil form. As part of the design process many variables are considered both with respect to the general field profile but also the manner in which the magnet will be used.
Proper design assures a robust winding while avoiding excessive
cooling losses due to excessive charging current or inadequate homogeneity. Superconducting magnets must operate below both the critical temperature and the critical field of the material from which they are constructed. Table 1 illustrates the critical temperatures and fields of the most common superconductive materials used to fabricate magnets. Figure1
shows how the capabilities of NbTi vary as a function of both temperature and field. For a superconducting magnet manufactured using NbTi, operation in
the superconducting state is only possible below the surface indicated in this 3 dimensional graph.
Fig1. NbTi superconducting properties.
So the limits on obtainable magnetic fields come from the materials used.