The Braess paradox which was originally identified and observed in road traffic context, shows that it may happen that by adding capacity to the network, the performance of all users degrades. This paradox has long been known in a framework called the Wardrop equilibrium, in which there are infinitely many individuals and in which the decision of one single individual has a negligible effect on the performance of the other individuals. Another framework in which such a paradox may occur is that of the Nash equilibrium in which there are a finite number of players, and in which the decision of each player has nonnegligible effect on the other players. It is natural to expect the same type of paradox in the Nash equilibrium context, whenever it occurs for the Wardrop equilibrium. In this paper, we present a case where a paradox similar to that of the Braess appears in a Nash equilibrium but does not appear in a Wardrop equilibrium in the same environment. We consider the model of load balancing in distributed computer systems. We further establish the uniqueness of the Nash equilibrium for this problem.