We present a comprehensive study of low temperature quantum transport in double gated InAs/GaSb composite quantum wells. Recently, it has been proposed that this system in inverted regime should exhibit the topologically insulating (TI) phase, characterized by an energy gap in the bulk and gapless edge modes, protected from backscattering by time reversal symmetry. We sweep the Fermi level through the bulk mini-gap, observing resistance peaks and finding strong evidence for the existence of the mini-gap; however, the mini-gap does not show insulating behavior, with a residual bulk conductivity which is a few times larger then the expected contribution from the edge. Our data indicate, that bulk conductivity is not an issue of "dirt", which can be improved by simply reducing the amount of disorder, but a fundamental property of strongly coupled electron-hole systems in realistic materials, which must be considered in studies of proposed TI edge modes.