Not necessarily.
Although a picture's colour depth may be 16 or 24 bit, the chances are it isn't actually using all the colours available in that colour depth. If it is, then many of the colours may be duplicated, or so similar that you wouldn't notice if they were adjusted to the same position in the palette.
If you use a decent paint package (I use Photoshop) then it will look at what colours are actually used, and then decide how best to reduce the colour depth.
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moto