Let me begin by saying that what I have written here is strictly my opinion. I do not claim my view is correct; I am sharing it because it is how I understand an age of accountability.
There are a couple of things which are necessary in order for a person to sin. First of all, obviously there must be a law to break. The Bible says, ". . . where no law is, there is (or there can be) no transgression" (Rom. 4:15). The Scripture adds that ". . . sin is not imputed when there is no law" (Rom. 5:13). However, from the very beginning of time, there was law. Adam and Eve disobeyed God's law which said they were not to eat of the tree of the knowledge of good and evil (Gen. 2:17).
Thirdly, the person must be capable of comprehending that he is choosing to deliberately defy God. A new born baby can not possibly sin. While he has the Adamic nature which makes him predisposed to sin, until he is old enough to comprehend it all, he remains innocent.
The passage of Scripture most people point to concerning what has come to be called "man's sin nature," is found in Romans 5:12, which says, "Wherefore, as by one man sin entered into the world, and death by sin; and so death passed upon all men, for that all have sinned." I would like to suggest what the Word is teaching is that it is our sin that results in our "inheritance" of Adam's consequences, not his. Notice it says death passed upon all men because all men sin!
Another passage, which I believe is misunderstood, is 1 Corinthians 15:22, which says, "For as in Adam all die, even so in Christ shall all be made alive." How does one get to be "in Christ?" We must believe what God says concerning His Son. Similarly, I believe one gets to be "in Adam" (in his likeness - Gen. 5:3; become part of fallen humanity) when he rejects what God says. In other words, I believe that until a child consciously chooses to do what he knows God forbids him to do, he is still pure in the eyes of God. Deuteronomy 1:39 says, "Moreover your little ones, which ye said should be a prey, and your children, which in that day had no knowledge between good and evil, they shall go in thither, and unto them will I give it, and they shall possess it."
The Holy Spirit gives conviction when He decides a child is ready!