Many of the observations of the static Jahn-Teller effect for orbital triplets in cubic symmetry correspond to tetragonal or trigonal distortions as predicted by linear coupling theories but in a number of cases orthorhombic distortions or coexisting inequivalent minima of the adiabatic potential energy are observed. In the present paper we investigate systematically a number of possible theoretical explanations for these observations. Assuming that the dominant terms in the vibronic Hamiltonian are the linear vibronic couplings to eg and t2g vibrational modes, we study the influence of many other effects within a perturbative approach. The minima of the adiabatic potential energy for a triplet in cubic symmetry are first calculated to lowest order in all the coefficients for quadratic nonlinear vibronic coupling and cubic anharmonicity. This is used to determine the conditions for orthorhombic or coexisting inequivalent minima to occur. Then a perturbative approach to the problem of multilevel coupling is developed and applied to the case of a T2g level vibronically coupled to another level. In lowest order, the effect of interlevel coupling is to introduce terms formally analogous to nonlinear vibronic coupling terms in the single-level problem. The conditions for having the various kinds of distortions are given and discussed in each case. The influence of induced linear vibronic coupling to another vibrational mode is then considered and it is shown that distortions other than tetragonal, trigonal, or orthorhombic may occur. Finally we calculate the exact eigenvalues of the vibronic Hamiltonian with spin-orbit coupling to a spin of (1/2), including the lowest-order nonlinear vibronic couplings and anharmonicities, and discuss the role of spin-orbit coupling.