Thanks everyone! Will take a look at all of this for a few days and hopefully not burn my eyes out 
I read a paper once where the researchers were mapping exactly 26 colors. They created an alphabet of maximally distinguishable colors, then encoded text in color squares and taught people to read. It worked, but they commented that 26 was at the outer limit of what was distinguishable, even under carefully controlled conditions with constant background color and illumination.
Unfortunately, I don’t have a citation.
I notice that the colors cmyk posted leave room for black and white backgrounds (i.e., allow 28 colors in all, if those background colors count).
Here’s a palette of 26 colors with black and white included.
Very important point. … And one reason I didn’t try too hard to tune the above palette: improving it on my monitor might make it worse on yours.