Yes you are correct. Take something like the Super Nintendo, it typically displayed hundreds of colors on screen, however each graphical element could have a maximum of 16 colors (4 bits).
The way they solved this was by having CLUTs (color lookup tables), so for the Super Nintendo (if my memory serves me) you had 16 different CLUTs that you could assign to a graphical object, each able to contain 16 unique colors.
This is what made something like the Super Nintendo graphics look so much better than say the Amiga 500, both used bitmap graphics, but the Amiga did not have a CLUT solution, so you were stuck with 32 colors except for more advanced graphic modes which weren't really usable for games.
The way they solved this was by having CLUTs (color lookup tables), so for the Super Nintendo (if my memory serves me) you had 16 different CLUTs that you could assign to a graphical object, each able to contain 16 unique colors.
This is what made something like the Super Nintendo graphics look so much better than say the Amiga 500, both used bitmap graphics, but the Amiga did not have a CLUT solution, so you were stuck with 32 colors except for more advanced graphic modes which weren't really usable for games.