So I had to cross look up your verse, because you have not actually posted the chapter and verse, because I figured context would explain a lot. And it DOES!!
First, it is Matt 5. That is the start of the Sermon on the Mount, where Jesus is preaching the Beatitudes. Wonderful teaching on the Kingdom of God.
DO NOT THINK I HAVE COME TO ABOLISH THE LAW OR THE PROPHETS, BUT TO FULFILL THEM!
“Do not think that I have come to abolish the Law or the Prophets; I have not come to abolish them but to fulfill them.[SUP]18 [/SUP]For truly, I say to you, until heaven and earth pass away, not an iota, not a dot, will pass from the Law until all is accomplished. [SUP]19 [/SUP]Therefore whoever relaxes one of the least of these commandments and teaches others to do the same will be called least in the kingdom of heaven, but whoever does them and teaches them will be called great in the kingdom of heaven. [SUP]20 [/SUP]For I tell you, unless your righteousness exceeds that of the scribes and Pharisees, you will never enter the kingdom of heaven." Matt 5:17-20 ESV
Jesus is saying that the law does not save us, because we cannot possibly keep the laws, and neither could the scribes and Pharisees. I am not downgrading the importance of obeying God in all things, but Jesus clearly says in Verse 20, that if we think keeping the law makes us righteous, then we better be very careful that our law keeping exceeds that of the scribes and Pharisees, or we will never enter the Kingdom of God (Heaven is a Jewish euphemism for God in this phrase!)
Further, the word "abolish" or "relaxes" in Greek, is actually the word λύσω, which really means to loose or destroy. (It is the first verb you really parse, because it follows the rules perfectly in every tense!) So Jesus is saying he did not come to completely destroy the instructions or commandments (ἐντολῶν), but rather to bring a better way for us to not be constantly separated by sin from God, except for the sacrificial system. He fulfills the commandments because of who he is, and what he is going to do on the cross, not because of what we do! (Although I think if we all followed the words of the Beatitudes, and the 10 instructions of God found in Exodus 20:1-17 and repeated in Deut. 5:7-21, what a better world we would all have!)
Verse 20 comes with a huge qualifier of verse 5:19 that we simply cannot earn our way into the Kingdom of God!
The rest of the Sermon in chapter 5 of Matthew continues with all the ways we cannot obey the law, including anger, lust, divorce, oaths, and retaliation.
Then comes the key verses which we need to harken to. Love your neighbor, your enemies, and pray for those that persecute you.
"You have heard that it was said, ‘You shall love your neighbor and hate your enemy.’ [SUP]44 [/SUP]But I say to you, Love your enemies and pray for those who persecute you, [SUP]45 [/SUP]so that you may be sons of your Father who is in heaven. For he makes his sun rise on the evil and on the good, and sends rain on the just and on the unjust." Matt 5:43-45
Chapter 6 deals with:
Giving to the Needy (a big issue in both the Old Covenant and the New Covenant!)
The Lord's Prayer
Fasting in secret
Laying Up treasures in heaven
Not being anxious but trusting in God
Chapter 7 deals with:
Not judging others
Ask and it will be given
The Golden Rule
A Tree and its Fruits
I never knew you (referring to those who have done mighty works, but never known the Lord: a warning for the Prosperity
Gospel!)
Build your house on the Rock
And finally, Jesus ends the Sermon on the Mount and people were astonished at this teaching - because he was teaching as one who had authority.
"[SUP]28 [/SUP]And when Jesus finished these sayings, the crowds were astonished at his teaching, [SUP]29 [/SUP]for he was teaching them as one who had authority, and not as their scribes." Matt 7:28
Following Jesus is never about us keeping the law, but rather following the words and the person of Jesus Christ. He is the authority, and he is the way, the truth and the life.
And when you pull a verse like Matt 5:19, which comes from such an important part of the Bible - Jesus public teaching to the ordinary people, you need to look at the context of the whole Sermon, which is three chapters long!!
And do notice, that other than divorce (adultery) really, the whole Sermon is fresh and exciting. It is not about keeping 613 Old Testament laws, or the Sabbath (I think if it was that important, Jesus would have mentioned it here!) or any other Old Testament laws, nor about Feast Days or how to worship God (with or without instruments??)
It is about new laws, written on our hearts by the Holy Spirit, when we walk with Christ!
“This is the covenant that I will make with them
after those days, declares the Lord:
I will put my laws on their hearts,
and write them on their minds,” Hebrews 10:16
Getting back to the title of the OP, this is why keeping the commandments are not enough to make you a Christian!