Design of optimized shimming equipment is under way for UHF MRI of the human brain. Here shimming theoretical limits are explored. With a simple model, proof is given that, depending on field source distribution around a zone of interest, no shimming hardware external to such a zone can fully mitigate the inhomogeneous field. A simulation is performed on a 100-subject database to establish hard shim limits on the whole brain. On the other hand, 3D region-specific shimming is shown to be a very effective way to improve homogeneity in critical zones such as the pre-frontal cortex and around ear canals.