Triplet excitation energies have been calculated for Ne, H2O, HF, BH, N-2, and C-2 using the full configuration interaction (FCI) model and the coupled-cluster model hierarchy CCS, CC2, CCSD, CC3, and CCSDT, where CCS, CCSD, and CCSDT are the standard coupled-cluster models where singles, doubles and triples are successively added and where CC2 and CC3 are approximations to the CCSD and CCSDT models where approximations are introduced in the highest amplitude equations. Comparing the coupled-cluster excitation energies with the FCI results shows that the excitation energies are improved at each level in the hierarchy up to CC3. The CC3 and CCSDT excitation energies have similar accuracy for the single excitation dominated excitation energies, whereas the double excitation dominated excitation energies are significantly improved also from CC3 to CCSDT. Singlet excitation energies have also been calculated for HF using the coupled-cluster hierarchy up to CCSDT. Triplet and singlet excitation energies with similar single excitation weights have about the same accuracy.