The 170Er(p,t)168Er and 166Er(t,p)168Er reactions have been studied using 18-MeV protons and 17-MeV tritons, respectively. The reaction products were analyzed with magnetic spectrographs providing overall resolutions ranging from 10 to 15 keV full width at half maximum. The chief aim of these studies was to search for previously unknown 0+ states in 168Er, and one new 0+ level, at 1833.5 keV, has been identified. All the known excited 0+ states were populated much more strongly in the (t,p) reaction than in the (p,t) reaction. Numerous natural parity states in known bands were also observed in both reactions. The significance of these results for some recently proposed interpretations of 168Er is discussed. The states with I,Kπ = 0,0+ at 1422 keV and 2,2+ at 1848 keV are both populated much more strongly in the (t,p) reaction than the single-phonon γ vibration. Thus the interpretation given these levels in the interacting boson description, as predominantly multiple-phonon excitations involving at least one γ phonon, does not satisfactorily describe these data. The quasiparticle-phonon model of Soloviev and co-workers is the only model that does not conflict in some way with the presently available experimental data.