Is it possible to be a Christian by birth? My parents are both Christians. When we were kids they always took us to church. Does that make me a Christian?
No.

God knows each one of us as a person, not just part of a family. This is clear even in the Old Testament:


  Ezekiel 33:20 (ESV)
  Yet you say, 'The way of the Lord is not just.' O house of Israel, I will judge each of you according to his ways."


Looking at the Gospel in the New Testament, we see that again each of us is an individual to God:


  John 3:16 (ESV)
  "For God so loved the world, that he gave his only Son, that whoever believes in him should not perish but have eternal life.


God saves whoever believes in Jesus from perishing. Only by believing, yourself, can you be saved.