During regulated protein degradation, the 26S proteasome recognizes ubiquitinated substrates through its 19S particle and then degrades them in its 20S enzymatic core. Despite this close interdependency between proteasome subunits, we demonstrate that knockouts from different proteasome subcomplexes result in distinct cellular phenotypes. In particular, depletion of 19S PSMD lid proteins, but not that of other proteasome subunits, prevents bipolar spindle assembly during mitosis. Despite decreased ubiquitin-mediated protein degradation in PSMD knockouts, we find that the monopolar spindle phenotype is instead caused by the aberrant degradation of the kinesin motor protein KIF11. We show that KIF11 degradation occurs through the 20S proteasome in a ubiquitin-independent manner upon loss of 19S proteins and that the resulting alterations in spindle forces lead to the unique monopolar phenotype. Thus, the presence of the 19S particle ensures proper spindle formation by restraining ubiquitin-independent degradation.