Antenna bounds are a useful tool in assessing the feasibility or performance of an antenna design. Microstrip patch antennas are often limited by their relatively narrow bandwidth, and therefore Q-factor is an important design parameter, as it is related to the inverse of the fractional bandwidth. This article presents the first tight lower Q-factor bounds on microstrip patch antennas supported by an infinite dielectric substrate. The derived lower Q-factor bounds are orders of magnitude tighter than the Chu limit and introduce a new scaling rule. These bounds consider all possible geometries on the predefined design region. Moreover, it is shown that well-known patch antennas have Q-factors near the bounds and have thus a near-optimal bandwidth. The computation of the bounds is done using a method of moments (MoM) formulation. However, an approximation to these bounds using commonly available simulation tools is provided.