We feature our recent work in the field of single-molecule magnets (SMMs) using inelastic neutron scattering (INS). The term "pressure" in the title has a triple meaning. First, there is the expectation from research-funding agencies and the public to make some significant steps towards applications. Second, the synthesis of new compounds and the applications of physical techniques for their understanding are being pushed to their limits. And third, by applying hydrostatic pressure, valuable insight into the mechanisms behind the SMM phenomena can be gained. Examples from our research have been taken to illustrate these points. After a brief introduction to the technique, the strength of INS for the accurate determination of exchange and anisotropy interactions in SMMs is highlighted. We hope to demonstrate that, by pushing INS measurements to their limits, i.e., using the best available neutron sources and instrumentation, combined with high quality samples, new insights into the relevant physical processes in SMMs can be gained.