The question has been pretty well covered, but I’ll just add a bit:
1.) We can certasinly see the colors in the spectrum. There’s no reason to think there are any gaps in this regime. In fact, the roble,m is the opposite – several regions of the CIE Chromaticty diagram are perceived as the same color. We have a problem with resolution. Dave MacAdam did a lot of work on this, and the regions of confusion are called MacAdam Ellipses, after him.
2.) We can see from blue up through red on the spectrum, and some people can maybe see a bit beyond (especially if they’ve has artificial lenses implanted, I understand – they transmit a bit further than your natural lens) With very bright sources (like lasers) you can see a bit beyond the range you normally see, since otherwise the response there is normally too low. But I’m told that Really Far Red looks just like the furthest Red you normally see – your eye/brain system doesn’t see it as a different color. In other words, it falls into the same Macadam ellipse.
3.) Purple, on the other hand, lying along the line connecting the ends of the spectral locus on the Chromaticity Diagram, does appear to be a different color. Not in the same ellipse. To me, Indigo and Vioollet look more like darker blues, and purple is something else.
4.) Pink is definitely in the Chromatocity diagram – it lies along a line between the “white” center and a point in the red on the spectral locus. exactly where it lies depens upon how “red” it is, and how light it seems to be.
5.) Brown, on the other hand, is an interesting concept, and I confess that I have to look into it further. Since it’s made up of a mixture of colors, it ought to have a location on the Chromaticity Diagram. I’m tempted to say that a lot of browns are really very dark yellows (especially since my colonoscopy, which got to watch on color TV), but I find that answer not completely satisfying. Needs further study.
6.) certainly, people haven’t seen “all colors” if you mean visually perceiving all wavelengths of electromagnetic wavelengths, or even the EM wavelengths that can be perceived by living animals. Bees and, I think, horseshoe crabs can see further into the UV (and see it as a distinct color, unlike human viewers of UV). Pit Vipers can detect infrared. And the Mantis Shrimp has extraordinarily broad visual sensitivity.
7.) What about sensitivity to other properties of light? Bees and Horseshoe Crabs can detect polarized light. People have a very limited ability to do this (google “Haidinger’s Brush”). Some varities of Mantis Shrimp, it’s recently been learned, can detect circularly polarized light. The way they perceive the world must be vastly differrent from the way we do. Does “left hand circularly polarized red light” count as a different color from “right hand circularly polarized light”? How about to a Mantris Shrimp? Is it different from “Horizontally Linearly Polarized Light”?