Core-shell nanorods (NRs) with InGaN/GaN quantum wells (QWs) are promising for monolithic white light-emitting diodes and multicolor displays. Such applications, however, are still a challenge because intensity of red band is too weak as compared with blue and green ones. To clarify the problem, we have performed power and temperature dependent, as well as time-resolved measurements of photoluminescence (PL) in NRs of different In content and diameter. These studies have shown that the dominant PL bands originate from nonpolar and semipolar QWs, while a broad yellow-red band arises mostly from defects in the GaN core. Intensity of red emission from the polar QWs at the NR tip is fatally small. Our calculation of electromagnetic field distribution inside the NRs shows a low density of photon states in the tip that suppresses the red radiation. We suggest a design of hybrid NRs, in which polar QWs, located inside the GaN core, are pumped by UV-blue radiation of nonpolar QWs. Possibilities of radiative recombination rate enhancement by means of the Purcell effect are discussed.