In the past unusual features in the excitation spectrum of the S = I compound CsNiCl3 have been attributed to remaining Haldane correlations in the 3D ordered state. One of these features is the degenerate triplet excitation at the 1D AF zone center, which is split by the magnetic field. Recently a higher-order spin-wave expansion has been proposed, in order to explain the experiment without assuming the Haldane correlations which are typical for integer spins. We studied CsNiCl3, and CsMnI3, the S = a counterparts of CsNiCl3,and demonstrate striking differences. The energies and polarizations of CsMnI3 are well explained by linear spin-wave theory. The experimental findings on CsNiCl3, however, disagree with the higher-order spin-wave theory, while a theory of Affleck which takes into account Haldane correlations correctly models the qualitative behaviour. Hence we have shown that the degeneracy of three modes at the 1D AF zone center of CsNiCl3, and their strong held dependence, are caused by Haldane correlations which are only present in integer-spin systems.