The mounting hardware used on the cargo boxes allows the cross bar to sit almost flush with the bottom of the box, gap is approximately 1/8".
When determining overall height of the box above the cross bar or roof keep in mind that not all cross bars are flat. Some factory and aftermarket bars have material on top of the bars that will increase this gap. Many factory bars will bow upwards in the center which will increase the height of the box.
If you are trying to determine if a given box will fit into a garage we would recommend adding between a 1/4 and 1/2" to the outside height dimensions provide here.
Tower and bar heights for both round and aero style bars
|Tower||Height (Base to top of bar)|