The Bose-Hubbard Hamiltonian describes the competition between superfluidity and Mott insulating behavior at zero temperature and commensurate filling as the strength of the on-site repulsion is varied. Gapped insulating phases also occur at noninteger densities as a consequence of longer ranged repulsive interactions. In this paper we explore the formation of gapped phases in coupled chains due instead to anisotropies tx≠ty in the bosonic hopping, extending the work of Crepin et al. [Phys. Rev. B 84, 054517 (2011)., 10.1103/PhysRevB.84.054517] on two coupled chains, where a gap was shown to occur at half filling for arbitrarily small interchain hopping ty. Our main result is that, unlike the two-leg chains, for three- and four-leg chains, a charge gap requires a finite nonzero critical ty to open. However, these finite values are surprisingly small, well below the analogous values required for a fermionic band gap to open.