Arguably the most-famous quantum phase transition is the Bose-Einstein condensation (BEC) of magnons. It can be studied experimentally in gapped quantum paramagnets when an external field drives down the gap via the Zeeman effect until it closes at a critical field. For the BEC universality class to apply, at least axial symmetry is required. The organic quantum magnet DTN has attracted particular attention in this context as its tetragonal crystal structure holds promise to the required axial symmetry. Field-induced ordering in DTN has been studied extensively and has been interpreted in the context of BEC of magnons. It is therefore puzzling that a large gap is found to open in the 'BEC' phase by ESR and neutron scattering which is so far not fully explained. Here we seek to explore the origins of this gap using high-resolution inelastic neutron scattering.