Let me first bring to remembrance our oft-stated claim that we adhere to the Bible, not to human-made tradition. It's important to keep in mind that even a "firm" doctrine which we've known and believed all our lives may turn out to be unBiblical. When that comes to light, we have to make a decision to retain the tradition anyway, or to let it go in favor of the Biblical view.
We've had a tradition, for centuries and centuries, that the Bible is divided into two sections: the Old Testament and the New. We even support this two-fold division by quoting 2 Tim 2:15, which says, "Study ..., rightly dividing the word of truth."
But this two-fold division is not the division known by the early church, and presented by Jesus. We know from historical/secular records that the Jews of Jesus' day divided their scriptures into a three-fold division, that of the Law (Torah), the Prophets (Nevi'im), and the Writings (Kethuvim) (or the "Tanahk", a word formed (with vowels added) by taking the first letter of each of these divisions, TNK).
When the church was established, a fourth division was added to the Holy Scriptures, that being the inspired writings of the early church.
Jesus refers to the three-fold division of the Jewish scriptures, as well as making an oblique reference to the fourth, in Luke 24:44, where he mentions:
1. the Law
2. the Prophets
3. the Psalms (representing the Writings)
4. "my words"
Why does this matter?
1) Because that's the Biblical division. This should in and of itself be sufficient as a reason to recognize a four-fold division of the Bible rather than a two-fold division.
2) Because by lumping the three-fold division of the Jewish scriptures into a single unit, we then misunderstand the teachings concerning the "old covenant".
The term "old covenant" is referenced in 2 Corinthians 3:14, where it is equated in the next verse to "Moses", and thus by implication to the Mosaic Covenant, aka the Law, aka the Torah. To extend this term to also refer to "the Prophets" and "the Writings" is to go beyond the text.
The term "old covenant" is also referenced in Hebrews 8 and 9 where it speaks of the ritual components of the Torah of Moses. To extend the term to also include "the Prophets" and "the Writings" is to go beyond the text. This term is also explained by the writer of Hebrews as the covenant referred to by Jeremiah (see next point).
Jeremiah defined the covenant which would be replaced by the new covenant, in Jer 31:31ff. In this passage, Jeremiah makes it clear that the old covenant which would be replaced by the new is the covenant which YHWH made with the Israelites at the time He brought them out of Egypt, aka the Mosaic Covenant.
This means that the 39 books we know today as the "Old Testament" (aka TNK) have not been replaced by the New Testament. Only the T portion of the TNK (or more specifically, the Mosaic Covenant portion of the T portion) has been replaced by the new covenant. Or in other words, the portions of the Tanahk which are not part of the Mosaic covenant have not been replaced by the New Covenant, but are in fact, still in effect.
We see this is true by logic: we accept the rainbow of the Noahic covenant as still being in effect, trusting that God will not flood the earth again as he did in Noah's day. (If this covenant with the inhabitants of Earth is still in effect, we are still to multiply and be fruitful, and we should still execute murderers.)
We see this is true by the doctrine of Jesus, who based his teachings concerning marriage, divorce, and remarriage on the validity of the Creation account. If that text is no longer in effect, then Jesus' teachings about marriage are no longer in effect; he did not create a new doctrine out of his own authority, but used the authority of the Creation text. Do away with that authority, and you do away with Jesus' doctrine.
We see this is true by the doctrine of Paul, who explicitly says in Galatians 3 that we are part of the Abrahamic covenant, and that the Mosaic covenant was added in as a temporary covenant, but that this temporary covenant did not nullify the first covenant. It was this temporary Mosaic covenant which was replaced by the new covenant.
We see this is true by the example of the Beroeans in Acts 17:10ff, who turned to the Tanakh as their authority in measuring the words of the apostles, and were praised as noble for doing so.
We need to stop teaching people that the Bible is divided into two portions, the man-made division of the Old Testament and the New Testament. Instead, we need to make it clear that the old covenant which was replaced by the new was the Mosaic covenant, not the 39 books we refer to as "the Old Testament".