Depends heavily on the type of game. If the game requires quick reactions; such as a fighting game or a a hack and slash, a high FPS rate is crucial. One of a myriad of reasons why the original release of the DmC reboot is such a step down from previous Devil May Cry games in my opinion is the 30 FPS frame rate; not only is your reaction speed cut in half, but the game just doesn't look as smooth, which is a big contributor to the feeling of free flowing movement.
Stable 30 FPS is acceptable (though not preferable) for most titles, but have been way too many recent titles with issues reaching even that. Arkham Knight on PC, the original Dark Souls, Dead Rising 3, Ryse, Titanfall, etc. Going anywhere below 30 severely hampers player reaction and compromises immersion.