Most knights were either of low noble background, or knighted after making their fortunes. Most also had done governmental or military service to the crown; the modern practice of recognizing the arts with a knighthood simply didn't exist.

Francis Drake, for instance, was already a successful privateer in the service of the crown before he became a knight. Walter Raleigh had first been introduced at court as a child, but wasn't knighted until after he helped suppress Irish uprisings against England. As an arbitrary example of a non-military knight made by Elizabeth I, Nicholas Mosley started off his career as a weaver of wool cloth, but was knighted only after serving as Lord Mayor of the City of London.

Shakespeare was never in government, never a military officer, and poor. Knighting him wouldn't have made any sense to the monarchs of his day.