There are several well-known games in which the pareto optima and Nash equilibria are disjoint sets.
The most famous is probably the prisoner's dilemma.  Races to the bottom or tragedies of the commons typically have this feature as well.

I proposed calling these inefficient games.  More generally, games where the sets of pareto optima and Nash equilibria are distinct (but not disjoint), such as a stag hunt could be called potentially inefficient games.

It seems worthwhile to study (potentially) inefficient games as a class and see what can be discovered about them, but I don't know of any such work (pointers welcome!)