But even if you're fully vested, if you can't sell your shares, they're essentially worthless (technically the term is probably "illiquid asset"). Until there's an "event" (IPO, acquisition, probably more), they can't be turned into real money.
If you exercise and sell at the same time, you will pay short-term income taxes, but without any AMT to worry about.
I agree that exercising and selling can be a good strategy but we're (mostly) talking about private companies here where that may not be an option due to a lack of liquidity.
http://www.startupcompanylawyer.com/2009/01/11/should-a-comp...
That would be a Big Red Flag for me.