This Bible passage has the same numbers for chapter and verse as e.g. John 3.16, that can be said to be a kind of abstract of the Bible's Christian message. Since the numbers happen to coincide, this verse also gets a hightened glow for some Bible teachers. Be that as it may - 2Tim 3.16 is used in ways that often become problematic.
In claiming that this verse teaches the authority of the Bible, two errors are made, one logical and the other chronological.
A. Circular reasoning
"The Bible claims that the Bible is authoritative. We can believe this, since the Bible is authoritative, which we know since the Bible claims it."
This seems to be an (often unspoken) line of thought with many who quote the verse we are inspecting. When this line of thought is written down in this way, it is easy to see that it is an example of circular reasoning, i.e. that the conclusions are used to prove the postulates. But since it is seldom written down, it is easy to hide this logical somersault.
In most cases, this is probably not done with ill will, but rather through thoughtlessness, based on poor Bible teaching. It is, however, surprising how many otherwise intelligent people fall in this trap.
B. The meaning of the word, "Scripture"
When Paul wrote to Timothy, the anthology we know as the Bible didn't yet exist. Parts of it weren't even written, and the very idea of a "New Testament" hadn't been broached. The books of the Old Testament had been written, but the final table of contents hadn't been decided on as yet.
So, when Paul uses the terms, "the sacred writings" and "scripture", in vv. 15 and 16 respectively, we must look to the context to see what he means. It cannot be The Holy Writ in today's meaning of the expression. Through his mother and grandmother, both of whom were of Jewish descent, Timothy had been taught the Scriptures since childhood. Isn't it obvious that Paul is referring to the Jewish Scriptures, that a few decades later were crystallized into our OT?
In my opinion, therefore, 2Tim 3.16 cannot be used to prove that the Bible is divinely inspired. Such a use of the verse would twist it into a shape that violates its actual message.
This is not to say, however, that the Bible isn't divinely inspired, just that other arguments are needed to prove the idea.
I wish to make an additional comment:
Paul writes in the previous verse that point of Scripture is "to instruct you for salvation through faith in Christ Jesus." Scripture - whether a proto-OT, as here, or the present-day Bible - is not meant as a school book in science, law, history or sociology, just to mention some examples, but its point is, rather, to help us to Christ. Nor is it the knowledge gained from the Bible that saves, but faith in Christ.
I wanted to say this, because I've seen too many examples of Bible abuse and bibliolatry, i.e. Bible worship. It could even be claimed that the Bible is the Christians' most common idol - a trap that we by all means possible must try to avoid!