Do Calvinists believe that it is possible to be saved without believing the doctrine of predestination? From a Calvinist theological view, is it possible for God to predestine that XYZ be saved, but for XYZ to reject the doctrine of predestination throughout his or her life?
Without any question any person who never believes in predestination can still be considered a saved and regenerate predestined Christian by a Calvinist.  All that is required to be considered 'predestined' in the eyes of men, is that a person believes in Christ for the forgiveness of their sins.  Anyone who truly 'confesses Christ' in this sense will be saved, regardless of how clear, or confused they may be on the surrounding doctrines of their salvation.


  If you declare with your mouth, “Jesus is Lord,” and believe in your heart that God raised him from the dead, you will be saved. 10 For it is with your heart that you believe and are justified, and it is with your mouth that you profess your faith and are saved. 11 As Scripture says, “Anyone who believes in him will never be put to shame.” 12 For there is no difference between Jew and Gentile —the same Lord is Lord of all and richly blesses all who call on him, 13 for, “Everyone who calls on the name of the Lord will be saved.” (NIV Romans 10:9-13)


Knowing how Calvinists leaders have thought through history, I can assure you if someone added a requirement like 'faith in the doctrine of predestination' to be needed in addition to 'faith in Christ' they would have denounced it. It is a 'person' who died for our sin that we must put our faith in, not a textbook statement phrased by church leaders.