SWe consider systems of ultracold spin-zero neutral Bosonic atoms harmonically trapped and loaded in an optical lattice. We consider physical situations such that the gap between the different energy eigenstates along a selected spatial direction is much smaller than all remaining ones. This setting realizes a generalized multi-band Bose-Hubbard model. We obtain a full characterization of the zero and finite temperature transitions to superfluidity, showing that in the ordered phase the superfluid is distributed among the different bands. This situation provides a realistic possibility to realize fragmentation of superfluidity in physical systems.