Although late stage gap formation reduces the surface density in the vicinity of protoplanets, simulations suggest gas may continue to leak through the protoplanets tidal barrier, replenishing the gas supply and allowing protoplanets to acquire masses comparable to or larger than that of Jupiter. Global gas depletion is a possible explanation for gaseous planets with lower masses in weak-line T-Tauri disks and ice giants in our own solar system, but it is unlikely to have stalled the growth of multiple systems around nearby stars that contain relatively low-mass, close-in planets along with more massive and longer period companions. Here, we suggest a potential solution. We show that supersonic infall of surrounding gas onto a protoplanet is only possible interior to both its Bondi and Roche radii. Although the initial Bondi radius is much smaller than its Roche radius, the former overtakes the latter during its growth. Thereafter, a positive pressure gradient is required to induce the gas to enter the Roche lobe of the protoplanet and flow is significantly reduced. We present the results of analysis and numerical simulations to show that the accretion rate increases rapidly with the ratio of the protoplanets Roche to Bondi radii. Based on these results we suggest that in regions with low geometric aspect ratios gas accretion is quenched, resulting in relatively low protoplanetary masses.