Did Shakespeare Have Any Rivals When He Was Alive?

Shakespeare did have one rival, who was born in the same year as him, 1564, but unlike Shakespeare who lived until 1616, his 'rival' Christopher Marlowe died in 1593.

Had Marlowe lived for longer he may well have given Shakespeare a run for his money. Marlowe wrote plays such as Dr Faustus and the Jew of Malta. His works invoked the flesh (and sins thereof) and the Devil with a passion which had hitherto been unknown.
He also wrote poems such as 'The Passionate Shepherd' and 'Hero and Leander', although these are not perhaps comparable to the Shakespearean sonnets.
He settled in London in 1587 and his works were extremely popular within the Elizabethian world.
His life was cut short in 1593 when he criticised the Bible and was suspected of being a heretic. Before he could be taken before the Privy Council to try him on this charge he was found dead. Some people allege this was due to a drunken fight, others that it was due to men such as Walter Raleigh wishing to conceal information Marlowe had. However, the end result was that at the age of 29, just as Shakespeare was coming into his glory. Marlowe died.

One 'rival', may well have been Robert Greene. Greene, educated at both Oxford and Cambridge universities, married well, and, with everything before him, decided to lead a life of debauchery, drink and hell-raising, until, at the age of just 32, he died in the gutter, unloved, unwanted and uncared for. One of his last acts was to pillory Shakespeare, in a jealous attack, ostensibly telling his university colleagues, ( Christopher Marlowe, Thomas Nashe, George Peele, etc) to beware, "An upstart crow! Who, beautified with our feathers, supposes he is as well able to bombast out a blank verse as the best of you, and with his 'Player's hide 'rapt in a Tigers skin', is in his own conceit, the only Shake-scene in a country!"