The first "non-Biblical" argument to be presented is that the Bible itslef was ultimately determined by tradition and the Church. There were in existence plenty of letters and books which could have just as easily been used in the Bible but weren't (for whatever reason). I'm not referring here to the so-called "Agnostic Gospels" such as the "Gospel of Thomas," which make completely outrageous claims about any manner of subjects (for example, not allowing women into heaven). I referring to such writings as the Book of Enoch (which was referenced by St. Paul and a number of the Church Fathers such as St. Augustine). Also, the Church could have just as easily disregarded some of the other books which were eventually accepted into the canon. The Book of Revelations is an excellent example of this.
Ultimately, the books that were canonized were those which had been a part of the "canon" all along, e.g. those books which were traditionally a part of the cannon. Even considering Luther's rejection of the Deuterocanonicals (among other things), he did not seek to add anything new to the Bible, thus demonstrating that he accepted at least that part of the Church's tradition. That no books were later added by the Lutherans or any other Christian denominations is especially telling here.
Another thing that people tend to lose track of when discussing Sola Scriptura is that the New Testament letters were just that: letters. They were written for the explicit purpose of being presented to their stated recipients (Romans, Corinthinas, Colossians, Timothy, etc.). They most certainly were not written "to the people of the world at large." In fact, by most accounts the letters weren't all collected together and composed into a sort of book until decades (at least) after being initially written. In any case, it was not until about the middle of the second century that a sense of a canon began to arise... and parts of the current canon weren't even canonized until the fourth century (source, albeit a harsh one)!
This presents and interesting dilemna for the Sola Sciptura apologists out there. The Sola Scripturalist, in claiming that God's revelation to us was complete with the Bible must therefore necessarily claim that it was complete with the writing of the last single part of the Bible, namely the last letter of the New Testament. However, the New Testament itself wasn't actually put together until much later. The question arises, what should the early Christians rely on before this time? The answer is simple: tradition and the authority of the Church.
After all, if St. Paul could write a letter to the Colossians which would later become authoratative for all Christians, would he not also have the authority to preach to the Romans that which he wrote to the Colossians? And if he had happened to be pass through Colosse at a later time and find that they had not heeded his letter, would he not have had the authority to correct them in person? Or if a missionary from Colosse had found that the Galatians were facing the same problems as had the Colossians before, would he not be within reason to try to help fix the problem?
But here I begin to digress away from the main point of this post, which is not so much authority as tradition. The concept of the authority of the Church is a whole new post in and of itself; not too mention, I know that many of you will disagree more strongly with me on that point than the one which i am presently discussing. In any case, there's more to be said about tradition.
Perhaps most importantly is that tradition gives us a good double-check for interpreting scripture. After all, a theology based on scipture would fail us if we had interpreted the entire thing incorrectly. We all know at least somebody who "knows" their Bible inside and out, but whose interpretation thereof leaves something to be desired. The existence of tradition gives us something to compare our interpretations against. If the interpretation clearly contradicts traditional beliefs which the Bible is not attempting to condemn, then there may be something wrong with the interpretation.
Furthermore, tradition gives an at least indirect context within which to interpret scripture. Much of the Bible is clearly meant to be interpretted within some sort of cultural context. For example, must of Christ's ministries tied back to everyday life in the greater "Palestian" region. Taken out of the cultural context they become a bit difficult to apply to our lives, don't you think?
Finally, there are the problems that are caused by using Sola Scriptura as a literal means of living. Consider, for example, the increasing problem of people misinterpreting the writings of St. Paul. In these, he states on occasion that marriage ought to be delayed. In the correct cultural and historical (not to mention traditional) context, he is generally referring to either a) celibacy (as in the "celibate service" generally associated today with the Catholic clergy) or b) delaying marriage during times of famine or persecution (in which case starting a family would generally be a bad idea).
Increasingly prevalent today however is the mindset that singleness itself is a gift to be sought after. As Debbia Maken of "Boundless" argues, this is not the case.
"On the whole of history, past generations of Christians saw singles under a divine obligation — one might say a duty — to marry.... The laws and practices of these former cultures likewise conveyed to all what was normative and what behavior was expected."Yet with greater frequency than ever we see people misinterpreting St. Paul's Epistles in such a way as to believe that singleness itself ought to be glorified. Worse still,
"Instead of placing this modern phenomenon of protracted singleness under Scripture for scrutiny, we have done the exact opposite — we have made Scripture the handmaiden to the phenomenon."This seems to be becoming more widespread, too. All one has to do is observe the increasing number of girls involved with our local "Campus Crusade for Christ" who are "taking a year (or more!) off, for Christ." It should also be noted that when none of the guys willingly followed suit with this plan, but rather tried to dissuade their female friends, the girls by-and-large became indignant and tried to offer a "scriptural" defense of their decisions.
The result: the "Cru" guys largely look elsewhere (which may at times include non-Christian circles) to find dates. And this is but one example of what can go wrong with throwing out tradition when interpreting the Bible. In other words, tradition is necessary to prevent the sometimes disasterous consequences of misinterpretating scripture.
The moral of the story is, don't throw out tradition as a part of an attempt to "get back to basics." It just doesn't work.