We show that there exist non-linearly realised duality symmetries that are independent of the standard supergravity global symmetries, and which provide active spectrum-generating symmetries for the fundamental BPS solitons. The additional ingredient, in any space-time dimension, is a single scaling transformation that allows one to map between BPS solitons with different masses. Without the inclusion of this additional transformation, which is a symmetry of the classical equations of motion, but not the action, it is not possible to find a spectrum-generating symmetry. The necessity of including this scaling transformation highlights the vulnerability of duality multiplets to quantum anomalies. We argue that fundamental BPS solitons may be immune to this threat.