Follow-up question: I often read that Shakespeare "invented" hundreds or thousands of new words. Is this figure remotely accurate? Is it possible that some of these words were common in spoken English and Shakespeare was the first to use them in print?

Shakespeare did have a large influence on the language, partially due to being born in the right place at the right time. Before Shakespeare's lifetime, most books printed in Europe were in Latin. During his life, England alone printed thousands of items in English, and literacy rates were rising rapidly.

The Renaissance as a whole added at least 10,000 new words to the English lexicon. As writing in the vernacular became more popular, authors borrowed words--mostly from Latin, French, Spanish, or Dutch-- to fill the holes they saw in English. But how many of these words can we credit directly to Shakespeare? The Oxford English Dictionary claimed at least 2,000 English words were first used by the playwright. But this is in need of revision, as many of these words were actually used previously by a lesser-known author. The OED is currently being revised, a process that will take many years. But from the pages edited so far, we can conclude that around 30% of words credited to Shakespeare were actually first used by someone else.

But the true genius of William Shakespeare lies in the way he put words together, and created new metaphorical definitions for existing words. The updated OED currently claims that Shakespeare invented around 8,000 new word senses. Here is a good quote by Bernard Levin that shows many of the phrases we use today were invented by Shakespeare.

souces: The Story of English and Bad English