His contemporaries at the time appear to have harboured suspicions.
Playwrights putting their name to other people's works and taking the credit for other authors' words was a persistent problem in the days before copyright and intellectual property laws. One specific missive written by Robert Greene in 1591 complaining of the practice appears to refer to someone with similarities to Stratford's Shakespeare.
In 1591, dramatist Robert Greene wrote a missive complaining of playwrights putting their name to other people’s work and other “underhanded brokery”. He began warning others in the industry about an “upstart Crow” in 1592, who thinks he is the “onely Shake-scene in a countery”. Many literary historians believe Greene this Crow was Shakespeare.
Shakespeare’s achievements, going from a glover’s son to Britain’s most prominent playwright, would have been a cause for anger and jealousy. His rise was a slap in the face to those better off with access to more elite education and more opportunities. It is no surprise, therefore, that his genius would have put him in the crosshairs of his contemporaries. This could explain Greene's malicious missive. There are also several rivals who concede Shakespeare’s genius and leave glowing reviews. Ben Jonson, Shakespeare’s great literary rival coined the term the “Swan of the Avon”. He references the great playwright’s talent, then places him as being from Stratford-upon-Avon. It is inconceivable that he would do that to his rival if Shakespeare was merely a front for another author.
[P1] There were rumours in Shakespeare's life that he did not write the plays. [P2] Therefore, it is possible that he did not author the plays.
[Rejecting P1] Rumour is not sufficient ground to question his authorship, particularly given Shakespeare would have been a source of envy for his rivals.