Hi Michael and Michael!
Fortin is correct in that the margin for error grows as you use larger and larger test designs. Huge tests can be done successfully. It just gets harder.... You will need
massive amounts of traffic and a highly controlled test environment to make it work.
Take a look at
http://www.york.ac.uk/depts/maths/tables/orthogonal.htm for a few possible Taguchi designs. You will see the L81 which uses *40* variables. Not really practical for online testing, but it is out there.
Personally, I think even 12 variables is too many for most websites. Particularly since most people don't really know how to design a test effectively to control noise and maximize the strength of their success metrics.