Not my HD3850 either. Not sure which cards support 32 bit Z-buffers to be honest. Maybe some workstation cards like Quadro/FireGL?
The Z-buffer contains the depth value (Z-axis) of each pixel on screen. It's used along with depth testing to make sure you don't draw a triangle in the back of the scene over the top of a triangle in the front of the scene. (geometry is rarely drawn in order back-to-front) The number of bits used in the depth/Z buffer determines the number of distinct depth values each pixel can have. The more bits, the less chance of triangles appearing in the "wrong order" (often seen with chalk lines flickering in LFS).
