How is the New Covenant consistent with Deuteronomy 4:2? The King James Bible; Deuteronomy 4:2, says:


  Ye shall not add unto the word which I command you, neither shall ye diminish ought from it, that ye may keep the commandments of the LORD your God which I command you.


How does this make sense with the New Testament? How can there be a New set of laws given when the Old Testament clearly says not to add on?
The simple answer is that the New Testament neither adds nor subtracts anything from the Old Testament, the New Testament is the more perfect and fulfilled Word of God. For example, John 8:3-11.

And early in the morning he came again into the temple, and all the people came unto him; and he sat down, and taught them. And the scribes and Pharisees brought unto him a woman taken in adultery; and when they had set her in the midst, They say unto him, Master, this woman was taken in adultery, in the very act. Now Moses in the law commanded us, that such should be stoned: but what sayest thou? This they said, tempting him, that they might have to accuse him. But Jesus stooped down, and with his finger wrote on the ground, as though he heard them not. So when they continued asking him, he lifted up himself, and said unto them, He that is without sin among you, let him first cast a stone at her. And again he stooped down, and wrote on the ground. And they which heard it, being convicted by their own conscience, went out one by one, beginning at the eldest, even unto the last: and Jesus was left alone, and the woman standing in the midst. When Jesus had lifted up himself, and saw none but the woman, he said unto her, Woman, where are those thine accusers? hath no man condemned thee? She said, No man, Lord. And Jesus said unto her, Neither do I condemn thee: go, and sin no more.

Notice that Jesus said "Neither do I condemn thee, go and sin no more." rather than "Thou hast no guilt, wrongly accused were thee". Of course, there are very defined things that contrast Judaism and Christianity, things that were changed by Jesus, such as the view of the Sabbath day and clean and unclean foods. 

It must be taken into account that the verse shown (Deuteronomy 4:2) says "YE shall not add unto the word which I command you, neither shall YE diminish ought from it...". This meaning that the Israelites, and mankind as a whole for that matter, shall not add unto nor diminish from the holy word of God, and not that he is not allowed to. God the father said for hundreds of years by means of prophets, that the day would come when the Messiah would come to earth and make a new and improved covenant with Jew and Gentile alike. That is what He did through His Son, Jesus. Jesus was a representative of the father and embodied that new and wonderful Covenant (John 10:30, Matthew 3:17, Romans 1:4, etc.).