Not really, or at least, it's misleading to say so. Every time you increase the colour depth you're using more space - an extra bit for every pixel in every image, each time.
However, as the amount of space you use increases linearly, the number of colours you can use goes up exponentially:
1 bit: 2 colours
2 bit: 4 colours
3 bit: 8 colours...
Basically, the difference between two colours and four colours is the same increase in memory as the difference between 32 and 64, and so on for larger powers of two.