The coupled-cluster method with single, double and triple excitations (CCSDT or SUB3) is applied to a system with up to six electrons outside a closed shell. The Ne atom and its ions serve as a test case. Starting from Ne6+, electrons are added one at a time all the way to Ne. Ionization potentials and excitation energies in the 2p shell are calculated. CCSDT gives good results for all sectors, with an average absolute error of 0.11 eV, whereas methods including triple excitations approximately or not at all fail for the higher sectors.