The structure of Pr skutterudites provides large cages for the anharmonic rattling motion of the rare-earth guest atom. For suitable composition, their frequency may be nearly degenerate with low-energy crystalline electric-field excitations of localized 4$f$ states. In this case, the anharmonic oscillation of Pr is supplemented by nonadiabatic or vibronic effects due to 4$f$-electron--phonon coupling. This affects the temperature dependence of the effective phonon frequency and the rattling phonon-Raman contribution to NMR and nuclear quadrupole resonance relaxation. Furthermore, dispersive 4$f$-quadrupole excitations originating from intersite quadrupole coupling may exhibit typical anomalies due to mixing with the phonons. A model of anharmonic rattling phonons together with dispersive 4$f$ excitations is introduced to study these effects. It is of particular relevance to Pr(Os${}_{1\ensuremath{-}x}$Ru${}_{x}$)${}_{4}$Sb${}_{12}$, where the degeneracy of 4$f$ excitations and rattling phonons has been identified before.