Is it possible for a believer to stop believing?
It is conceivable because of the hardening that can take place in one's heart because of sin (see Heb. 3:7ff), a believer could reach a place in the downward spiral of rebellion and disobedience when he or she no longer really believes in anything. But can a true believer reach a state where he or she could be classified as a stark unbeliever? Some because of their view of the meaning of apostacy would say yes, but most who hold to the historic doctrines of conservative Christianity would say an emphatic no to the idea that a believer could ever become an unbeliever.
When we speak of a believer, we are talking about a person who has trusted in the person of Jesus as the One who died for our sin and rose again, and having recognized their sinfulness and separation from God, has trusted in Christ alone for their salvation.