Well actually William Shakespeare MAY have been born with a different name, because there is quite a lot of argument about the way his name is actually spelt.
While we have definitive ways of spelling words nowadays, in Elizabethan time spelling was more flexible and even Shakespeare spelt the same word in different ways in his plays
In fact they make a "joke" about this at the begining of the film Shakespeare in Love where Shakespeare himself is seen signing his name on a piece of paper but spelling it many different ways.
You can read more about the spelling of his name here
http://shakespeareauthorship.com/name1.html