One of the biggest 'problems' for the Bible is its seeming acceptance of slavery. The Old Testament seems to have different voices on the issue. At points it goes as far as prohibiting Israelites from enslaving other Israelites (Lev. 25:39-43). At other points it shows an understanding of the status and value of slaves that is no different than that of other Ancient Near Eastern nations (Ex. 21:28-32). The New Testament is radical at points, suggesting that in Christ there is no slave or free (Gal. 3:28), but at other points it seems to reinforce the status quo (Ep. 6:5-9).
Ephesians 6:5-9 comes at the end of a section called 'the household code,' where Paul regulates the life of the Christian household. Marriage is dealt with at length, first. Second is parent-child relations, with slave-master relations being third. In each of these issues, Paul reinforces the cultural norms of the time, submission from the member lower in the hierarchy to the member higher in the hierarchy (see O'Brien p. 402). While not endorsing or promoting slavery, this text, and others like it, certainly doesn't condemn it. Why? Aren't slave/free distinctions erased in Christ? In Philemon, doesn't Paul urge Philemon to free Onesimus? Is Paul an inconsistent thinker?
In addition to these challenges, there is a desire (albeit a very noble one), to claim that Philemon shows that institutional slavery is wrong. It does no such thing. In Philemon, Paul deals with the issue of a single relationship between master and slave. The same Paul wrote Colossians and Ephesians as well, so he clearly didn't make the leap that some interpreters want to make. Philemon is silent on the issue of institutional slavery. With that said, I do believe that we can argue from the Bible against slavery, but I would suggest arguing from a different text, Genesis 1:26-27, that human beings are created in the image of God, and thus to subjugate another human being, whether through slavery or other means of repression, is a sinful violation of a fellow image bearer and an implicit denial of that truth. And closely related I believe that, using the model Richard Hays developed, of looking at ethics through the lens of the concept of 'new creation,' applied not only to Christians but to the redemption of all of creation also causes us to move away from slavery and towards support of freedom. 
But we still have a 'problem. Why don't we see clear movement in the New Testament towards the abolition of the institution of slavery? The best answer I saw in the commentaries was from Doug Moo (p. 370-378). He suggests several possibilities that together help explain the phenomenon which I will paraphrase below:
- Slavery was so much a part of the world in its day that, as an institution, it would almost have escaped the notice of early Christians (p. 371).
- 'Freedom' or 'liberation,' was not the obvious good that it is to us in the modern world. It was not racially based nor were all slaves of low economic status, some raised their status through slavery (p. 371).
- The Christian movement was too small to make a difference and more importantly, the category of social action did not exist as they were not in a democracy, but an empire (pp. 371-2).
- Earthly realities were less important than eternal spiritual realities, however this is not to suggest that the transformation of earthly realities was unimportant (pp. 372-3).
What do we do, is the 'problem' solvable? I think that we need to remember that the purpose of the Bible isn't to directly answer all of life's questions. God wants us to think hard about how to live out his Word in the way most consistent with what he has revealed in it within our contemporary cultural context. I would suggest that fighting for the abolition of slavery is more consistent with Scripture in our context than being permissive of slavery's continuance. We must draw the conclusion that Paul and other New Testament authors did not draw.
On a related note, the issue of slavery in the New Testament shows us, in a crystal clear manner, that applying Scripture isn't a matter of finding a proof text for the issue we're investigating. Applying God's word is hard work and in some rare cases we me even need to cut against the grain of seemingly clear implications of our proof texts if they conflict with the outworking of the central tenants of our faith. The human authors of Scripture were not omniscient and did not always apply God's revelation in the same way they might have if the Bible were written to 21st century American society. The Bible was, and this is not a bad thing, a product of its cultural situation. Thus, while initially Moo's suggestions as to why the New Testament does not condemn slavery may not seem completely satisfying, they are. Why should we demand that New Testament authors answer the question of slavery with a modern answer, especially when the question we are asking does not parallel the situation of the New Testament world in many ways?
 Hays model suggests using a set of lenses to look through when formulating Christian ethics: 'cross,' 'community,' and 'new creation.' While Hays work is open to critique, his work on New Testament ethics is widely regarded as one of the best in its field.