Isaac Asimov tackled this argument, at one point. He noted that Shakespeare made mistakes that a more educated man would likely not have — the pendulum clock in Roman times, IIRC, a prime example. I believe he also discusses the array of likely suspects.

I am not expert enough to naysay his final judgement, which was that the man who wrote Shakespeare’s plays was, in fact, William Shakespeare…

Those of you interested in Alternate History may find Harry Turtledove’s Ruled Britannia of interest. It postulates an England that failed to stop the Spanish Armada, and has endured years of Spanish rule. Ring in Shakespeare, Queen Lizzie, and the rest to tell an interesting and entertaining, if unlikely*, story…

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*I say unlikely because, as I understand it, they’ve managed to find some of those Spanish galleons on the sea floor, and it’s remarkable how little a chance they had of winning. The Spanish had close to zero logistical skills, and many of the ships had numerous cannonballs which would not have fit into the cannons they had on board. The whole Spanish Armada was ill-prepared for any kind of real sea battle.

A link mentioned here suggests that poetic language as employed by Shakespeare, Wordsworth, and more recent writers such as Philip Larkin and Ted Hughes stimulates areas of the brain which are not reached by the same thoughts translated into more pedestrian language.

Stuart’s post also notes that expensive private schools like Andover are now doing their best to ensure their students don’t derive unfair advantage from such mental improvements…