The main argument that Shakespeare could not have written the
plays is that only a well educated member of the aristocracy would
have had the time or the knowledge to write such fine literature.
This is the argument used in favour of both Edward de Vere and
Francis Bacon.
The same argument can be used to show that the Beatles did not
write their own music (in fact this argument was used during the
early 1960's); that Van Gogh never did any painting; and that Haydn
was unable to compose a symphony. (All of these people had as
little specialist education as Shakespeare had).
The argument also shows that another great playwright of the
Elizabethan era - Ben Jonson - couldn't have written his plays
either. (Despite his reputation as an intellectual, Jonson was as
uneducated as Shakespeare).
William Shakespeare wrote Shakespeare's plays - nobody else
could have.