Extension of the closed-shell coupled-cluster (CC) theory to studies of bond breaking and general quasidegenerate situations requires the inclusion of the connected triply and quadruply excited clusters, T-3 and T-4, respectively. Since the complete inclusion of these clusters is expensive, we explore the possibility of incorporating dominant T-3 and T-4 contributions by limiting them to active orbitals. We restrict T-3 and T-4 clusters to internal or internal and semi-internal components using arguments originating from the multireference formalism. A hierarchy of approximations to standard CCSDT (CC singles, doubles, and triples! and CCSDTQ (CC singles, doubles, triples, and quadruples! schemes, designated as the CCSDt and CCSDtq approaches, is proposed and tested using the H2O and HF molecules at displaced nuclear geometries and C-2 at the equilibrium geometry. It is demonstrated that the CCSDt and CCSDtq methods provide an excellent description of bond breaking and nondynamic correlation effects. Unlike perturbative CCSDT and CCSDTQ approaches, the CCSDt and CCSDtq approaches do not fail at large internuclear separations, in spite of using the restricted Hartree-Fock reference. All CCSDt and CCSDtq approaches are essentially n(6) procedures and yet they are shown to provide reliable information about T-3 and T-4 components, whose standard evaluation requires expensive n(8) and n(10) steps.