The simple answer is: yes, no, sort of!
Groups of players worked in ensemble, often running more than one play at once and many, many different plays in quick succession. Look up the Lord Chamberlain's Men, later King's Men, to find out more about the system of patronage, licensees, and performance as related to Shakespeare specifically.
Say Shakespeare's group were performing two plays this week, one each night. It made sense, and still does, for the actors who played the largest parts with the most lines in play A to take less strenuous parts in Play B. On the other hand these men had remarkable memories, they had to have a good handful of plays ready to perform at a moments notice, so this was by no means always the case.
You might have heard of William Kempe, who allegedly danced from London to Norwich, he was one of Shakespeare's most popular actors. He was an accomplished comic and many of the Bard's greatest comic roles are supposed to be influenced by Kempe, including Falstaff.
Kempe's successor, Robert Armin, was also an accomplished fool, but of a different sort, so the characters that Shakespeare wrote with him in mind were the more solemn, wise fools like Touchstone and Lear's fool.
Richard Burbage was an acclaimed actor, incredibly talented and popular by all accounts, who played Shakespeare's (and other contemporary playwrights) greatest roles including Lear, Hamlet, Othello, and others.
So in some cases, yes, certain actors nearly always played a specific role, but in others they probably mixed it up quite a lot. For example Armin is thought to have played Iago, one of the great Shakespearean antagonists.