Undoubtedly, Jonson and Shakespeare knew each other personally. The King's Men - who, as mentioned in the "About Volpone" section, put on the first performance of Volpone in 1605 or 1606 - was the company which acted many of Shakespeare's plays and in which Shakespeare himself acted. Shakespeare is even thought to have acted in one of Jonson's plays, Every Man In His Humour.
Jonson's relationship with Shakespeare was primarily one of respectful competition. Though he was tapped to write a poem introducing Shakespeare's First Folio, Jonson had previously commented that Shakespeare "wanted art" and, in response to claims that Shakespeare never blotted, or erased, a line, retorted "Would he had blotted a thousand." Jonson also once remarked that Shakespeare knew "small Latine and less Greek."
However, Jonson's main quarrel with Shakespeare was not personal but rather stylistic. Whereas Jonson prided himself on following the classical rules of dramaturgy (see the Epistle and the Prologue), Shakespeare relished just the opposite - breaking the classical rules of dramaturgy. Still, Jonson respected Shakespeare's natural genius. In the second folio of his work, published after his death, Jonson was magnanimous in his remarks on Shakespeare: "I loved the man, and do honour his memory, on this side idolatry, as much as any."