Yep, you're not the first person to notice this. :) With today's high contrast ratios, 8 bits per component is not enough to make a smooth gradient without visible banding - unless dithering is used.
Using more than 8 bits per channel on a display is called "deep color" by display manufacturers. It isn't very widespread because of a chicken-and-egg problem. A deep color display is useless without a video card that can output deep color, and a game engine that supports rendering to deep color. Likewise, there is no point in a game engine or a video card that supports deep color without the display. So there is not much incentive for hardware manufacturers and game developers to add support for this technology, as from either end, there is no market to justify the cost of development.
Also, there are other ways to fix banding due to the limited 8-bit precision. As I mentioned earlier, game engines can use dithering to hide the banding.
(Picture of a cat with a 256-color palette, without and with dithering. Created by Wikipedia user Wapcaplet, used under the CC-By-SA 3.0 license.)
Adding a slight dither of ±0.5/255 before writing out the pixel value to the framebuffer is extremely effective at hiding banding on smooth gradients, and essentially unnoticeable. If you're in an HDR engine, you do this during the tonemapping stage.
Finally, as others have noted, texture compression can be a bigger source of banding-like artifacts in the image than the 8-bit precision is. This may be what's going on with the sky in that picture, although it's difficult to tell - it's got so much JPEG compression on it that any artifacts due to DXT compression are pretty much swamped.