Here’s another claim:
Around 1601, students in Cambridge put on a play called The Second Part of the Return from Parnassus, the third in a series of plays that satirized the London literary scene. In this play, two characters named "Kempe" and "Burbage" appear, representing the actors Will Kempe and Richard Burbage of the Chamberlain's Men. At one point Kempe says,
“Few of the university [men] pen plays well, they smell too much of that writer Ovid, and that writer Metamorphosis, and talk too much of Proserpina and Jupiter. Why, here's our fellow Shakespeare puts them all down, aye and Ben Jonson too. O that Ben Jonson is a pestilent fellow, he brought up Horace giving the poets a pill, but our fellow Shakespeare hath given him a purge that made him bewray his credit.
This passage establishes that the playwright Shakespeare was a fellow actor of Kempe and Burbage, contrasts him with the University-educated playwrights, and establishes him as a rival of Ben Jonson.
Another author even said that “In 1602 the famous comedian Will Kempe wrote 'Here's our Shakespeare puts them all down, aye, and Ben Jonson too”.
Response: The famous comedian Will Kempe DID NOT say or write that. As mentioned by the first Stratfordian the quote comes from The Return from Parnassus Part 2, Act 4.3.1753-1760. The play was the third in a series played at St. John's College, Cambridge, between 1598-1602. They are likely to have been written by two or more authors who were probably college members, and they were acted by college students. Will Kempe, the 'famous comedian' was one of the characters being portrayed and the full quote was making fun at his ignorance.
The character Will Kempe was made to look ignorant, first, because 'Metamorphosis' wasn’t a writer, but the title of a work by the classical writer Ovid. Secondly, he is made to look ignorant because he's supposedly admiring his fellow Shakespeare for NOT writing like the university playwrights who "smell too much of Ovid" when, in fact, Shakespeare's Venus and Adonis, which the authors admire, is based on a story from Ovid's Metamorphosis (and headed by a couplet from Ovid's Amores), and The Rape of Lucrece was based on Ovid's Fasti. So, since the Parnassus authors are likely college students that admired classical authors as well as Shakespeare who also obviously liked classical authors, and since the character Will Kempe is being shown to be ignorant of 'his fellow' Shakespeare's reliance on classical authors, he is also probably being poked fun at for not realizing that 'his fellow' (the actor Will Shakspere) did NOT actually write the works under the name of Shake-Speare. Other parts of the play show the authors poking fun at actors for "mouthing words that better wits have framed" and because "They purchase lands, and now Esquires are named" (clear references to the actor Shakespeare). In other words, Kemp's "fellow Shakespeare" embodies the joke that Shake-Speare (as opposed to the actor Shakspere) was not Kemp's fellow at all. Instead, he was one that actually did share the “faults” of the university playwrights (“smelling of Ovid”) because he was one of them.
This is a very abridged response. A fuller response is found in Cockburn’s book and elsewhere. This is also discussed in Bate’s recent book Soul of the Age.