Total Depravity is the Calvinist belief that sin corrupts every aspect of the human condition, up to and including our ability to accept God. As human beings, we are said to be incapable of coming to God of our own free will. He must force us to do so.
Although I can see why Calvinists might glean Total Depravity from the Scriptures, I have never understood how they might maintain it. Scripture undeniably tells us that mankind is depraved, with each and every one of us turning to our own way, but it also tells us we have free will. We are held accountable for every sin, including the ultimate rejection of Christ.
In Romans 1:18-20, we are clearly told that those who reject the truth in unrighteousness are "without excuse". This clearly and indisputably shows us that we can repent. Inability, after all, is the perfect excuse.
Although Scripture indisputably presents some doctrine of depravity, the idea that man has literally no ability to come to Christ unless God forces him does not seem to be it. Yes, we do have verses such as John 6:44, but this need not mean man is literally unable to come to Christ. Two things must be considered.
First, we must ask why can no one come to the Son? This is not necessarily a statement of ability, but also of desire. There are many things we can do physically, but not mentally. For example, you cannot take a bite out of your own arm. Physically, it is entirely possible. If your teeth are intact, your jaws are perfectly capable of tearing off large chunks of your own flesh. You cannot do it not because you are incapable, but because it's contrary to your nature.
In much the same way, Jesus is contrary to our nature as humans. He is light, we are dark. He is Holy, we are sinners. He is perfect, we are tarnished. We simply do not want to come to Him. Our excuses are lame, which is why we are said to be without excuse, but we cannot come to Christ unless we are drawn simply because why would we? We are resistant, as those who have gone our own way, to accepting the ways of God.
Second, we must ask how are we drawn? This, scripture indicates, is not by some supernatural goading, but by external circumstances. These are still brought about by God. God can show you the stars, or strike you with blindness. God can tell you the word, or cut off your hearing. God's sovereignty is absolute. Nevertheless, scripture indicates that what God shows you is what He uses to draw you. Faith, as His word says, comes by hearing, and hearing by the word of God (Romans 10:17).
Furthermore, we have the examples of Tyre, Sidon, and Sodom. These three cities, according to Christ Himself, will find judgement day more tolerable than, for example, Chorazin and Bethsaida (Matthew 11:21-22). Why? Because if Jesus had done His miracles there, they would have repented. Note, it doesn't say "the Father would have drawn them". The miracles would have been the method of drawing them.
There is no sensible dispute that the Bible presents some doctrine of depravity, but when scripture is taken as a whole, it clearly presents us with the idea that our will is the problem. We are unable to come to Christ not because we are physically prevented unless we happen to be a part of the elect, but because we simply don't want to come to Him. Inability is the world's most perfect excuse, yet scripture tells us those who reject Christ are without excuse. Therefore, we must conclude that Total Depravity, while not heretical, falls short of being sound doctrine.