There are various theories that the plays attributed to Shakespeare were actually written by somebody else, such as the Earl of Oxford. I think one of the stronger arguments against this claim is the considerable lack of great literature, or great art in general, produced by members of the aristocracy.
I don’t believe that great art is associated with suffering. Everybody suffers in their life, no matter how well off they are–unrequited love, unexpected death, an unsightly pimple before the school dance. Great artists are not the people who suffer greatly, they are ones who can turn their suffering, or their imagination, into great art.
However, I do believe that great artists, at least prolific great artists, have immediate needs and long term aspiration. The pressure to produce something is an important element. Wealthy people are far less likely to have that pressure.
This is different from science, where, at least in the past, it was possible to make real discoveries as a dilettante. Several notable scientists were aristocrats. And of course there are exceptions. Lord Dunsany wrote quite a lot, and while there is only one Shakespeare, some of Dunsany’s work remains in print today.
Still, I think it far more likely that Shakespeare was a member of the middle class of his day, and I think it very unlikely that he was a member of the aristocracy.