The Floquet-Bloch theory for a periodic self-adjoint operator allows one to compute the spectrum as a union of discrete spectra of the Bloch Hamiltonian for quasimomenta in the Brillouin zone. In many practical cases one can, in fact, obtain the correct spectrum by running over only the boundary of the reduced Brillouin zone (in other words, the edges of the spectrum occur at symmetry points of the zone). In dimensions 2 and higher there seems to be no analytical reason for why this should be the case, yet counterexamples are difficult to come by or are numerically unconvincing. In collaboration with J. Harrison, P. Kuchment, and A. Sobolev we find examples for which the spectral edges are strictly and unambiguously found in the interior of the Brillouin zone - firmly resolving this question. The examples are first constructed for the graph and quantum graph cases and then bootstrapped to the higher dimensions.