  Observations have suggested substantial departures from pressure equilibrium
in the interstellar medium (ISM) in the plane of the Galaxy, even on scales
under 50 pc. Nevertheless, multi-phase models of the ISM assume at least
locally isobaric gas. The pressure then determines the density reached by gas
cooling to stable thermal equilibrium. We use numerical models of the
magnetized ISM to examine the consequences of supernova driving for
interstellar pressures. In this paper we examine a (200 pc)^3 periodic domain
threaded by magnetic fields. Individual parcels of gas at different pressures
reach widely varying points on the thermal equilibrium curve: no unique set of
phases is found, but rather a dynamically-determined continuum of densities and
temperatures. A substantial fraction of the gas remains entirely out of thermal
equilibrium. Our results appear consistent with observations of interstellar
pressures. They also suggest that the high pressures observed in molecular
clouds may be due to ram pressures in addition to gravitational forces. Much of
the gas in our model lies far from equipartition between thermal and magnetic
pressures, with ratios ranging from 0.1 to $10^4$ and ratios of uniform to
fluctuating magnetic field of 0.5--1. Our models show broad pressure
probability distribution functions with log-normal functional forms produced by
both shocks and rarefaction waves, rather than power-law distributions produced
by isolated supernova remnants. The width of the distribution can be described
quantitatively by a formula derived from the work of Padoan, Nordlund, & Jones
(1997).
