We explore two widely used algorithms for fluid reservoirs in molecular simulations and demonstrate that they may induce non-physical non-equilibrium effects, even in systems that should be at equilibrium. For example, correlations of momentum and density fluctuations lead to a bias in the mean fluid velocity when measured as the mean over samples of instantaneous fluid velocity. The non-physical behavior is entirely computational in origin and is an instance of a more general issue in molecular simulations: a failure to correctly model stochastic properties may induce non-equilibrium behavior that does not exist in the corresponding physical system. Finally, we demonstrate that simple algorithm corrections eliminate this artifact.