Yes, a Christian can celebrate birthdays. There is no biblical admonition to not have a celebration of one's birth. Still, some Christians do not want to celebrate birthdays because they feel that it might be a form of idol worship and they do not want to take away from the admiration of God. If this is their conviction, then they are obligated to adhere to that conviction.
But, millions of godly Christians celebrate birthdays on a regular basis. Of course, this does not mean that because so many do it that it is all right (argumentum ad populum 1). We must always look to the Scriptures for our final answer (sola scriptura2).
As long as a birthday is simply a celebration of one's birth and not an idolatrous occasion that robs God of his glory, then there shouldn't be a problem. After all, we are not under the law but under grace (Romans 6:14).