I was asked to explain what color and palette remapping capabilities EGA had and thought I would make a thread about it. EGA isn't well-understood these days because of VGA's success, but it's worth going over for people wondering why you can't set custom colors in all EGA modes.
EGA was introduced in 1984 with (limited) compatibility with existing IBM CGA monitors: 200-line graphics modes (320x200, 640x200) could use any of the existing 16 text colors per pixel. Only the high-res 640x350 mode could pick additional colors from a fixed palette of 64.
Using this definition, let's establish capabilities. CGA (1981) had 6 palettes, each holding 4 indexes, with each index pointing to one of the 16 text colors. Indexes could not be manipulated, so each palette is "fixed" and points to a set of 4 colors that cannot be changed.
The IBM PCjr (1983) (and, later, Tandy) expanded on CGA such that 16-color modes had a single palette with 16 indexes that could change what color each index mapped to. By default, indexes point to all 16 colors (#0 -> 0, #1 -> 1, etc.) but you could switch these around at will.
VGA (1987) is well-known but worth restating: In 256-color graphics mode, VGA has one palette holding 256 indexes, which each index able to reference one of 262,144 colors. The number 262,144 comes from 2^18, or 2^(3*6), as there are 3 color components (RGB), each 6 bits wide.
So what can EGA (1984) do? 200-line modes behave like PCjr: You can change which of the fixed 16 text colors each index points to. In the high-res 350-line mode, EGA behaves like VGA: You can reference one of 64 colors, or 2^(3*2), colors, as each component is only 2 bits wide.
For more info on programming EGA (as well as what came before and after it), two excellent references are Richard Wilton's "Programmer's Guide to PC Video Systems" and Richard Ferraro's "Programmer's Guide to the EGA and VGA Cards". Ferraro's book in particular the best for EGA.
