High-harmonic generation in solids by intense laser pulses provides a fascinating platform for studying material properties and ultra-fast electron dynamics, where its coherent character is a central aspect. Using the semiconductor Bloch equations, we uncover a mechanism suppressing the high harmonic spectrum arising from the coherent superposition of intra- vs. inter-band contributions. We provide evidence for the generality of this phenomenon by extensive numerical simulations exploring the parameter space in gapped systems with both linear dispersion, such as for massive Dirac Fermions, and with quadratic dispersion, as e.g. for bilayer graphene. Moreover, we demonstrate that, upon increasing dephasing, destructive interference between intra- and inter-band contributions is lifted. This leads to reentrant behavior of suppressed high harmonics, i.e. a crossover from the characteristic spectral "shoulder" to a slowly decaying signal involving much higher harmonics.