Hydrocarbon phase pressures at the peak of two severely overpressured reservoirs in the South Eugene Island 330 field, Gulf of Mexico, converge on the minimum principal stress of the top seal. We interpret that the system is dynamically constrained by the stress field present through either fault slip or hydraulic fracturing. In two fault blocks of a shallower, moderately overpressured reservoir sand, hydrocarbon phase pressures are within a range of critical pore pressure values for slip to occur on the bounding growth faults. We interpret that pore pressures in this system are also dynamically controlled. We introduce a dynamic capacity model to describe a critical reservoir pore pressure value that corresponds to either the sealing capacity of the fault against which the sand abuts or the pressure required to hydraulically fracture the overlying shale or fault. This critical pore pressure is a function of the state of stress in the overlying shale and the pore pressure in the sand. We require that the reservoir pore pressure at the top of the structure be greater than in the overlying shale. The four remaining reservoirs studied in the field exhibit reservoir pressures well below critical values for dynamic failure and are, therefore, considered static. All reservoirs that are dynamically constrained are characterized by short oil columns, whereas the reservoirs having static conditions have very long gas and oil columns.