The Observed Perlmutter-Riess Acceleration (OPRA) implies that the expansion of the Universe is currently increasing and is motivation for the so-called phantom energy models. We consider the dynamics of phantom scalar field models. An important physical time constraint, which can be used to rule out many cosmological models, is obtained from the condition that all forms of energy density, including the field causing OPRA (e.g., the phantom field), must be non-negligible for an extended period, which is conservatively estimated to be of the order of a few Gyr. We find that this physical time constraint cannot be satisfied in conventional phantom cosmological models.