The breakup of confined droplets was studied systematically for systems with either interfacial or component viscoelasticity. The former was obtained by adding a compatibilizer, the latter by using a viscoelastic fluid as the droplet or as the matrix phase. The critical capillary numbers of Newtonian and compatibilized droplets showed a similar increase with increasing confinement ratio. However, a decrease in breakup length was observed in the compatibilized case, caused by the viscoelastic interface. Viscoelastic droplets experienced more stabilization by confinement compared to Newtonian droplets. Matrix viscoelasticity, on the other hand, induced a destabilization with a minimum in critical capillary number as a function of confinement ratio, resulting from a complex interplay of viscoelastic stresses.