Is it possible for an unsaved person to believe that they are saved?

An unsaved person may believe they are bound for Heaven, but such a person would base their confidence on false views of self and of God. Specifically, they would believe themselves worthy of Heaven and they would believe God’s standard for entrance to Heaven is less than the Bible requires. Jesus spoke of such a person in the Gospels:

Luke 18:9  And He also told this parable to some people who trusted in themselves that they were righteous, and viewed others with contempt:
Luke 18:10 “Two men went up into the temple to pray, one a Pharisee and the other a tax collector.
Luke 18:11 “The Pharisee stood and was praying this to himself: ‘God, I thank You that I am not like other people: swindlers, unjust, adulterers, or even like this tax collector.
Luke 18:12 ‘I fast twice a week; I pay tithes of all that I get.’
Luke 18:13 “But the tax collector, standing some distance away, was even unwilling to lift up his eyes to heaven, but was beating his breast, saying, ‘God, be merciful to me, the sinner!’
Luke 18:14 “I tell you, this man went to his house justified rather than the other; for everyone who exalts himself will be humbled, but he who humbles himself will be exalted.”

Notice Luke introduces this parable as a story about a man who trusted himself rather than trusting in God. This is always the perspective of those who mistakenly think themselves bound for Heaven. The Pharisee saw himself as worthy of Heaven based on his own work. This is a perspective of the unbeliever who will not inherit the Kingdom of God. 

Meanwhile, the truly saved trusts in God’s mercy with the sacrifice of Jesus in his place. The saved recognize their sin before God and are thankful for the mercy of God in Christ. This person is saved by his faith in Christ rather than trusting in self.