No, a person can't be a real Christian and not know it. A true Christian is a follower of Jesus Christ, and even if they aren't perfect, they still have made a personal commitment to follow Him.
Your question, however, is very important because, unfortunately, the opposite can happen. It is entirely possible for someone to think they are a real Christian but not truly be one. Many people, in fact, say they believe in Jesus–and in one way they do. They believe He existed; they may even believe He died on the cross and rose again to give us eternal life. But nothing has happened to them inside, and they live just like everyone else.
What is wrong? The problem is that they've never personally committed themselves to Christ. To put it another way, they've failed to trust Him alone as their Lord and Savior. The Bible says, "Yet to all who received him, to those who believed in his name, he gave the right to become children of God" (John 1:12).
You see, the word "believe" in the Bible really means "to trust"–and that is what we all need to do with Christ. He has done everything necessary to make our salvation possible–but we must trust Him and commit our lives to Him. I pray you look at Christ and what He has done for you, and then trust Him alone for your salvation.