There has been much recent research interest towards understanding the phase behavior of colloidal systems interacting with a bridging attraction, where the small solvent particles and large solute colloidal particles can be reversibly associated with each other. These systems show interesting phase behavior compared to the more widely studied depletion attraction systems. Here, we use Baxter's two-component sticky hard sphere model with a Percus-Yevick closure to solve the Ornstein-Zernike equation and study the size effect on colloidal systems with bridging attractions. The spinodal decomposition regions, percolation transition boundaries and binodal regions are systematically investigated as a function of the relative size of the small solvent and large solute particles as well as the attraction strength between the small and large particles. In the phase space determined by the concentrations of small and large particles, the spinodal and binodal regions form isolated islands. The locations and shapes of the spinodal and binodal regions sensitively depend on the relative size of the small and large particles and the attraction strength between them. The percolation region shrinks by decreasing the size ratio, while the binodal region slightly expands with the decrease of the size ratio. Our results are very important in understanding the phase behavior for a bridging attraction colloidal system, a model system that provides insight into oppositely charged colloidal systems, protein phase behavior, and colloidal gelation mechanisms.