Thermodynamic stability of model single component and binary mixture fluids is considered with the Fisher-Ruelle (FR) stability criteria, which apply in the thermodynamic limit, and molecular dynamics (MD) simulation for finite periodic systems. Two soft-core potential forms are considered, phi(6,1)(r)=4[1/(a+r(6))(2)-1/(a+r(6))] and phi(2,3)(r)=4[1/(a+r(2))(6)-1/(a+r(2))(3)], where r is the separation between the particle centers. According to FR these are unstable in the thermodynamic limit if a>a(c)=1/2 and a>a(c)=(7/32)(1/3), respectively. MD simulations with single-component particles show, however, that this transition on typical simulation times is more gradual for finite periodic systems with variation in a on either side of a(c). For a<a(c), asymmetric density fluctuations are stabilized by the periodic boundary conditions. Also for binary mixtures of (stable) Lennard-Jones and phi(2,3) particles, phase separation into regions richer in one component than the other was observed for a<a(c). Binary systems with interactions similar to a model polymer-colloid fluid in the "depletion" limit equilibrated particularly slowly for a>a(c), with the unstable component in the mixture breaking up into many long-lived microdroplets which conferred apparent equilibrium thermodynamic behavior (i.e., negligible N-dependence of the average potential energy per particle) in this period.