Q:

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?

A:

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.

A:

No.
As Nicky Gumbel says on the Alpha course, "If you were born in a McDonalds, would that make you a hamburger?"
Furthermore, the Bible makes clear that we are all judged on our own standing with Jesus, e.g. Acts 2:21: "... everyone who calls on the name of the Lord will be saved."

A:

It depends what you mean by "Christian". The term has been used for millenia to refer to Christ's followers. However, one can follow Christ for a time, but never have truly believed in him, and eventually fall away. On the other hand, I typically associate "Christian" with someone who is truly saved. The term can mean different things to different people.
In fact, there have been some terms coined just for these types of discussions. One such distinction people have made is between the "invisible church" and the "visible church". The visible church is made up of, essentially, people who self-identify as Christians. Those whose parents are Christians and take them to church every Sunday would fall into this category.
I would even go further to argue that there are certain privileges afforded children of believing parents. These children may have the opportunity to hear the gospel and scripture on a regular basis. The may see modeled how a Christian marriage and family should look. The may learn how to pray. They may have the loving support of a church body.
However, these privileges are earthly privileges. 
On the other hand, is the invisible church. This is made up of all people who truly belong to the Lord, both living and dead, whether or not their parents are Christians, and whether or not they were in church this past Sunday. The privileges afforded these people is beyond compare.
One thing you will find made clear in the New Testament time and time again is that salvation is through faith alone in Christ alone.
In Jesus' day, it was thought that being a descendent of Jacob was sufficient for salvation. Paul rejects this explicitly at several points in the book of Romans. In Romans 2-3, he argues that Jews are not automatically saved, and again in Romans 9, he says "not all who are descended from Israel belong to Israel" and "This means that it is not the children of the flesh who are the children of God, but the children of the promise are counted as offspring." (ESV). It is clear, then, that it is your own faith, not that of your parents, that causes you to be numbered among the saints.

