What would it say other than "read the FAQ"?
How could it really be specific to the question enough to be useful? There are a million reasons a question can be downvoted. We'd have to be mind-readers to truly know why someone downvoted. (And if they really want you to know why, they'd leave a comment, if they feel like it.)
Also, the tooltip text on the voting arrows does provide a basic explanation, for the record; hover your mouse over the up and down vote arrows.
In my experience, types of users who tend to post questions that get downvoted aren't the ones willing to read much of anything at all.
I'd also argue that the burden of "why" always falls on the asker. If your question is downvoted, ask yourself how you can make it better. Prove that downvoter wrong by showing them how awesome and thorough and complete you can make your question!