Would William Shakespeare have had a "Brummie" accent?

Now don't get me wrong, I have nothing against accents. As someone that still has a Sussex accent after 40 years of living amongst Glaswegians I soon learnt that any "cracks" that came my way were meant in fun.

Shakespeare lived in the Birmingham area, he must have had an accent, so why do "Actors" all speak in what they call "A Shakespearean voice"?