We present the results of a detailed study of vertical surface brightness profiles of edge-on disk galaxies. Although the exponential disk scale height is constant to first order approximation, we show that for the large majority of galaxies in our sample, the scale height increases with distance along the major axis. The effect is strongest for early-type galaxies, where the increase of the scale height can be as much as a factor of 1.5 per scalelength, but is almost 0 for the latest-type galaxies. The effect can be understood if early-type disk galaxies have thick disks with both scale lengths and scale heights larger than those of the dominant disk component. Its origin appears to be linked to the processes that have formed the thick dis...