We report the existence of strongly localized modes in thin films and crystals with Fermi resonance interaction between vibration modes of molecules. Several families of bright and dark localized modes are investigated theoretically in two- and three-dimensional systems. Simple theoretical estimations are confirmed by numerical calculations. It is shown that in higher dimensions the solutions exist which do not have their one-dimensional analogs. The strongly localized Fermi resonance modes demonstrate the threshold behavior in any dimension in contrast to the nonlinear lattices with cubic interaction between sites.