The Bible does say that all the scriptures are inspired, which means "God-breathed":
All Scripture is God-breathed and is useful for teaching, rebuking, correcting and training in righteousness. (2 Timothy 3:16, NIV)
Though human writers did produce the scriptures, the ultimate source of the scriptures is God himself. When others spoke the scriptures they did so through God's breath.
In its original context this verse really refers only to the Old Testament. Peter however refers to Paul's letters as scripture, showing that the Apostles recognised that they were writing new inspired scriptures:
And remember, our Lord’s patience gives people time to be saved. This is what our beloved brother Paul also wrote to you with the wisdom God gave him— speaking of these things in all of his letters. Some of his comments are hard to understand, and those who are ignorant and unstable have twisted his letters to mean something quite different, just as they do with other parts of Scripture. And this will result in their destruction. (2 Peter 3:15-16, NLT)