So, long before the law were Adam and Eve. Just what do you think happened there? Or what about this:
Gen 4:6 The LORD said to Cain, “Why are you angry, and why has your face fallen?
Gen 4:7 If you do well, will you not be accepted? And
if you do not do well, sin is crouching at the door. Its desire is contrary to you, but you must rule over it.” (ESV)
Or:
Gen 39:8 But he refused and said to his master's wife, “Behold, because of me my master has no concern about anything in the house, and he has put everything that he has in my charge.
Gen 39:9 He is not greater in this house than I am, nor has he kept back anything from me except you, because you are his wife.
How then can I do this great wickedness and sin against God?” (ESV)
Or:
Gen 50:17 ‘Say to Joseph, “Please
forgive the transgression of your brothers and their sin, because they did evil to you.”’ And now, please forgive the transgression of the servants of the God of your father.” Joseph wept when they spoke to him. (ESV)
So, once again, we see that believers can indeed sin, since sin existed before the law and continues to exist afterward:
1Jn 1:8 If we say we have no sin, we deceive ourselves, and the truth is not in us.
1Jn 1:9 If we confess our sins, he is faithful and just to forgive us our sins and to cleanse us from all unrighteousness.
1Jn 1:10 If we say we have not sinned, we make him a liar, and his word is not in us.
1Jn 2:1 My little children, I am writing these things to you so that you may not sin. But if anyone does sin, we have an advocate with the Father, Jesus Christ the righteous. (ESV)
Also:
1Jn 5:16 If anyone sees his brother committing a sin not leading to death, he shall ask, and God will give him life—to those who commit sins that do not lead to death. There is sin that leads to death; I do not say that one should pray for that.
1Jn 5:17 All wrongdoing is sin, but there is sin that does not lead to death. (ESV)
The understanding then, of Romans 4:15, is that where prior to the law (since in context Paul is talking of the promise to Abraham), there is no knowledge of sin. More than that, you are taking that verse out of context. Look at what Paul writes right at the beginning of Romans:
Rom 1:18 For the wrath of God is revealed from heaven against all ungodliness and unrighteousness of men, who by their unrighteousness suppress the truth.
Rom 1:19 For what can be known about God is plain to them, because God has shown it to them.
Rom 1:20 For his invisible attributes, namely, his eternal power and divine nature, have been clearly perceived, ever since the creation of the world, in the things that have been made. So they are without excuse. (ESV)
So, even without the law of Moses, nature still tells us plainly about God, so that there is no excuse.