I’m sure the Crayola folks are working on that . . . 
Ahem! The “impossible colors” that that (paywalled) article is about are most certainly not ones that “every non-colorblind person knows […] they are real because they are all around us.” They are weird color experiences that can only be produced under very special conditions, in the laboratory (see [abstract] or [PDF]).
We agree, however, that there are many ordinary, common, and very real color experiences that cannot be explained purely in terms of the physics of light. The clearest example, to my mind, is the color brown, which we see all the time, but corresponds to no wavelength or mixture of wavelengths of light. You can’t make a beam of brown light, and you can’t experience your visual field as completely filled with brown - the nearest you will get is a dull orange or yellow, clearly distinct from brown - but brown is a real color, nonetheless. Pretending brown does not exist, or that it is “really” just a sort of orange or yellow, is the opposite of a true scientific attitude. (Strictly speaking, no colors can be fully and correctly explained just in terms of the physics of light, but, for some colors, physics can approximate an explanation that may seem satisfying, if you don’t think about it too hard.)
There is another type of impossible color. Since light that activates green cones will also activate either red or blue cones significantly, a “pure” green without blue or red is impossible. BUT, some researchers were able to stimulate green cones (chemically? – I can’t find a cite) enabling a subject to see a “psychedelic” color greener than green.
Interesting comments about brown. Here is an image in which the arrows point to a dark brown color and an orangish yellow. Those two very different-looking colors have the exact same RGB values!
Illusions like that one illustrates the problem pretty well, I think.
Basically, there are three levels to the color question. The first one is about physics: Wavelengths of light. The second one is about biology: The rods and cones in your eyes. Scientists have figured those out pretty well.
The third one is about the experience of color, and, no shocker here, this is where things get murky, because so much of it comes down to “your brain is making shit up to make sense of the world”.
A fascinating page on colors, in theory and in extremely well thought out tests, using the cameras and measuring devices in a way to explain to others as well as readers on color monitors; the man is an engineer of some sort, and is able to see beyond the normal spectrum after eye surgery, and wanted to figure it out: