"Blackbody" light sources like tungsten filament have a continuous spectrum, but differ in specific color temperature. But fluorescents, CFL's, and LED's all produce a discontinuous spectrum with spikes and imbalances; and therefore the best color renditions of those are actually quite expensive. CRI was a traditonal manner of expressing the statistically probability of how many of a hundred representative color patches would be properly rendered relative to human vision, according to any given lightsource. This was generally in context of opposing metamerism, where certain colors act chameleon-like under different sources. Anyone who aspires to balance a colorhead to a particular batch of color paper using a standard color reference should be acutely aware of this general principle.
But serious lighting engineers, along with pigment and dye experts, nowadays think more in terms of CIE color mapping and gamut properties, hoping to close at least a few of the loopholes. But nothing will ever be perfect. In fact, inkjet colorants are a bit of a step backwards in terms of gamut. No color film or RA4 paper is perfect either; every one of tehm has idiosyncrasies which need to be recognized.
I look at my own dried color prints not only with my expensive German color matching (5000K, CRI 98) tubes, but using several other sources too - diffuse daylight outdoors, high-end better-balanced LED spotlights, commercial fluorescent 5000K, warm tungsten, oddball CFL too if necessary. No different than what I did when our company paint color matchers came to me for a final assessment. Only half of it is physiological sight. The other half is psychological, with training and experience knowing what to look for, and how to compare, and in what light, according to the real-world conditions involved. One gets better at it over time.