I'm running 10.6.8 which is meant to be supported.
If you're trying to download the Cloud version of Illustrator (version CC) then you need OS X 10.7 or later. CC will not install or run on 10.6.8.
Illustrator CS6 will run on 10.6.8. That's a separate non-Cloud purchase although Cloud members do still have access to CS6 also.