A slim case can only be as slim as the height of the video card, unless it turns it 90 degrees. After that, it can only be as slim as the power supply is thick.
As nukemaster said, there are some great ITX cases that turn the video card sideways, but they use SFX PSUs to get even more slim.
I would personally go with a Silverstone ITX case + SFX PSU if I were building a slim machine.