My answer is "Yes", and a good example of it is found in Genesis 24 (yeah, I know, it's a book of fairytales to some, but you asked for my beliefs on the topic). Abraham's servant is sent to find Isaac a wife, goes to a well and prays, (to paraphrase) "Lord, may it be that the one You've chosen for Isaac will not only give me water to drink when I ask her, but she also offers to water my camels." So, along comes Rebekah, and, lo and behold, she does that very thing exactly.

Was Rebekah exercising free will? Absolutely. God didn't force her to do this. Was it predestined and preordained by the Almighty? Absolutely. So, somehow, free will and the sovereign will of God somehow work in perfect harmony. (Judas and Pharaoh are other good examples.) God isn't the Author of sin, but even when we screw up (deliberately or not), He can use it ultimately for His purposes to accomplish His will. That's what I am firmly convinced of because too many prophecies have come to pass precisely, even though the participants were acting totally free and independent.