One teaching that I heard when I was young as a Christian was this. God gives grace to the humble and He opposes the proud (Ja.4:6), if we are under grace sin cannot rule over us (Ro.6:14), and so, if we fall into sin, that means we are proud! Then what we need to do is to humble ourselves before God (Ja.4:10) so that He will be able to give us grace and sin will not rule over us. Simple? No, simplistic.
This is an example of a mechanical interpretation of the Bible. Another common example is to take Is.53:5 to mean that no sickness can affect those who believe in Jesus. What we need to do is to prepare our teachings in such a way that they fit together with all the other parts of the Bible in a cohesive way.
What Ro.6:14 means by being 'under grace' is that we are not under the old covenant where only those who obeyed all the Law could become acceptable to God (meaning nobody!), but under the new covenant of grace where our acceptance is a free gift God gives to us which we can never deserve. It doesn't mean that automatically sin will disappear from our life! In order to get victory when we face temptations, what we need to do is to avoid sin (put that sinful deed to death – Ro.8:13). We can see that this is not entirely up to God, but also to us. We may fail for various reasons – the temptation may take us by surprise, we didn't know what to do, our mind had not become sufficiently opposed to sin, we had indulged in this particular sin for a long time, we were already 'down' in our mind or feelings because of our circumstances, etc. If we don't give up when we fall, we can go to God for forgiveness (1Jn.2:1), ask God for help next time (He.4:16), learn about how to handle such temptations, etc. God will give us victory in an increasing measure.
As someone said, Ro.6:14 should be better understood as, "sin need not have dominion over you," rather than as "sin cannot have dominion over you." This would be in agreement with Ro.8:12 (we are not under obligation to live according to the flesh).
When I didn't know all this, whenever I fell into sin those days it drove me into condemnation, wondering where I had been proud, asking God to show me my pride, wondering if I was really 'under grace' after all, etc. But this was foolish and unnecessary. Now I know that grace is not something I qualify for by being humble and not having even an unconscious element of pride in me! When God accepts us, it is not only that we go to God as sinners (both in deed and also in being), but He also knows that He is dealing with someone who still battles with an old nature with its lustful desires (Ja.1:14). He knows that He is going to have to deal with the sinful nature that we are born with (Ps.51:5)) and even though He calls us saints, He has to deal with ungodly streaks that He finds in us, including the tendency to be proud.
Sanctification is progressive, as we discover our sinfulness, get cleansed and learn how to do better next time.