Core-shell nanorods (NRs) with InGaN/GaN quantum wells (QWs) are promising for monolithic white light-emitting diodes and multi-color displays. Such applications, however, are still a challenge because intensity of the red band is too weak compared with blue and green. To clarify this problem, we measured photoluminescence of different NRs, depending on power and temperature, as well as with time resolution. These studies have shown that dominant emission bands come from nonpolar and semipolar QWs, while a broad yellow-red band arises mainly from defects in the GaN core. An emission from polar QWs located at the NR tip is indistinguishable against the background of defect-related luminescence. Our calculations of electromagnetic field distribution inside the NRs show a low density of photon states at the tip, which additionally suppresses the radiation of polar QWs. We propose placing polar QWs inside a cylindrical part of the core, where the density of photon states is higher and the well area is much larger. Such a hybrid design, in which the excess of blue radiation from shell QWs is converted to red radiation in core wells, can help solve the urgent problem of red light for many applications of NRs.