We have investigated low-temperature electronic transport on InAs/GaSb double quantum wells, a system which promises to be electrically tunable from a normal to a topological insulator. Hall bars of $50,mu$m in length down to a few $mu$m gradually develop a pronounced resistance plateau near charge-neutrality, which comes along with distinct non-local transport along the edges. Plateau resistances are found to be above or below the quantized value expected for helical edge channels. We discuss these results based on the interplay between imperfect edges and residual local bulk conductivity.