Observations of X-ray binaries indicate substantial half-thicknesses for the accretion disks in these systems (up to h/R approximately 0.25, where h is the disk half-thickness and R its radius), while standard alpha accretion disks predict appreciably smaller half-thicknesses. We study the theoretical vertical structure of such disks using two independent numerical methods, and show that their maximum half-thicknesses in the subcritical regime cannot exceed h/R approximately 0.1. We consider various reasons for the apparent increase in the disk thickness, the most probable of which is the presence of matter above the disk in the form of a hot corona that scatters hard radiation from the central source and inner parts of the disk. As a result, the observed thickness of the disk and the illumination of its outer parts effectively increase. This mechanism can also explain both the optical-to-X-ray flux ratio in these systems and the observed parameters of eclipsing X-ray binaries.