  We study the behavior of systems in which the interaction contains a
long-range component that does not dominate the critical behavior. Such a
component is exemplified by the van der Waals force between molecules in a
simple liquid-vapor system. In the context of the mean spherical model with
periodic boundary conditions we are able to identify, for temperatures close
above $T_c$, finite-size contributions due to the {\it subleading} term in the
interaction that are {\it dominant} in this region decaying algebraically as a
function of $L$. This mechanism goes beyond the standard formulation of the
finite-size scaling but is to be expected in real physical systems. We also
discuss other ways in which critical point behavior is modified that are of
relevance for analysis of Monte Carlo simulations of such systems.
