I've thought this for years. Remember, when Paul wrote to Timothy that "All Scripture is inspired of God", Paul could only have been referring to what we know as the Old Testament. The New Testament writings were a work in progress at that time and were not at all considered Scripture. In fact, Luke's Gospel was a letter to Theophilus (Luke 1:3) and according to the "All Scripture" book, it was completed in 56 - 58 C.E. Paul's second letter to Timothy was completed in 65 C.E., just 7-9 years later.
I mentioned earlier in this thread that it was unclear to me if Luke quoted Paul or vice-versa. Well, now it seems clearer. Luke appears to have used Paul as a source. Paul completed the first letter to the Corinthians in 55 C.E., so the Gospel Luke was completed after that. Paul, who also was not an eyewitness of the Lord's evening meal, wrote what he wrote first, before Luke. Luke's words are very similar to Paul's words. (Compare 1 Cor 11:23 - 25 to Luke 22:19, 20)
It was only until the Bible canon was decided on that finally combined writings of the OT and NT to make up what we today consider Scripture. Just a thought.