Believe it or not, this is an important question. It's not simply a theological question. It's a theological question that has profound practical implications. Our answer will inevitably reveal our understanding of the gospel and reflect our understanding of sin and grace.
First things first: what total depravity isn't.
Total depravity does not mean "utter depravity." Utter depravity means that someone is as bad as he/she can possibly be. Thankfully, God's restraining grace keeps even the worst of us from being utterly depraved. The worst people who have ever lived could've been worse. So, don't read "utter depravity" into "total depravity."
Well, if total depravity isn't utter depravity, then what is it? As understood and articulated by theologians for centuries, the idea of "total depravity" means more than one thing.
On the one hand, total depravity affirms that we are all born "dead in our trespasses and sins" (Ephesians 2:1-3; Colossians 2:13), with no spiritual capacity to incline ourselves Godward. We do not come into this world spiritually neutral; we come into this world spiritually dead. Therefore, we need much more than to reach out from our spiritual hospital bed and take medicine that God offers. We need to be raised from death to life. In this sense, total depravity means we are "totally unable" to go to God. We will not because we cannot, and we cannot because we're dead.
None is righteous, no, not one; no one understands; no one seeks for God. All have turned aside; together they have become worthless; no one does good, not even one. (Romans 3:10-12)
For the mind that is set on the flesh is hostile to God, for it does not submit to God's law; indeed, it cannot. Those who are in the flesh cannot please God. (Romans 8:7-8)
Salvation only happens when God comes to us.
When the Resurrection and the Life says "Lazarus, come forth", the rest of the story does not depend on Lazarus. He can drag his feet all the way–admittedly, a hell of a thing to do–but he rises, no matter what. He just plain does… Jesus came to raise the dead. The only qualification for the gift of the Gospel is to be dead. You don't have to be smart. You don't have to be good. You don't have to be wise. You don't have to be wonderful. You just have to be dead. That's it. (Robert Capon)
So, in the sense above, Christians are obviously not totally depraved. We who were dead have been made alive.
But God, being rich in mercy, because of the great love with which he loved us, even when we were dead in our trespasses, made us alive together with Christ-by grace you have been saved-and raised us up with him and seated us with him in the heavenly places in Christ Jesus…(Ephesians 2:4-6)
But once God regenerates us by his Spirit, draws us to himself, unites us to Christ, raises us from the dead, and grants us status as adopted sons and daughters, is there any sense in which we can speak of Christian's being totally depraved?
Theologians speak of total depravity, not only in terms of "total inability" to come to God on our own because we're spiritually dead, but also in terms of sin's effect: sin corrupts us in the "totality" of our being. Our minds are affected by sin. Our hearts are affected by sin. Our wills are affected by sin. Our bodies are affected by sin. This is at the heart of Paul's internal struggle that he articulates in Romans 7:
For I do not do what I want, but I do the very thing I hate.
The painful struggle that Paul gives voice to arises from his condition as simul justus et peccator (simultaneously justified and sinful). He has been raised from the dead and is now alive to Christ, but remaining sin continues to plague him at every level and in every way.
Paul's testimony demonstrates that even after God saves us, there is no part of us that becomes sin free–we remain sinful and imperfect in all of our capacities, in the "totality" of our being. Even after God saves us, our thoughts, words, motives, deeds, and affections need the constant cleansing of Christ's blood and the forgiveness that comes our way for free. This is what J.C. Ryle was getting at when he wrote, "Even the best things we do have something in them to be pardoned."
While it is gloriously true for the Christian that there is nowhere Christ has not arrived by his Spirit, it is equally true that there is no part of any Christian in this life that is free of sin. Because of the totality of sins effect, therefore, we never outgrow our need for Christ's finished work on our behalf–we never graduate beyond our desperate need for Christ's righteousness and his strong and perfect blood-soaked plea "before the throne of God above."
The reason this is so important is because we will always be suspicious of grace ("yes grace, but…") until we realize our desperate need for it. Our dire need for God's grace doesn't get smaller after God saves us–in one sense, it actually gets bigger. Christian growth, says the Apostle Peter, is always "growth into grace", not away from it. Many Christians think that becoming sanctified means that we become stronger and stronger, more and more competent. And although we would never say it this way, we Christian's sometimes give the impression that sanctification is growth beyond our need for Jesus and his finished work for us: we needed Jesus a lot for justification; we need him less for sanctification.
The truth is, however, that Christian growth and progress involves coming to the realization of just how weak and incompetent we continue to be and how strong and competent Jesus continues to be for us. Spiritual maturity is not marked by our growing, independent fitness. Rather, it's marked by our growing dependence on Christ's fitness for us. Because we are daily sinners, we need God's daily distributions of free grace that come our way as a result of Christ's finished work. Christian growth involves believing and embracing the fact that, even as a Christian, you're worse than you think you are but that God's grace toward you in Christ is much bigger than you could ever imagine.
Because of total depravity, you and I were desperate for God's grace before we were saved. Because of total depravity, you and I remain desperate for God's grace even after we're saved.
Thankfully, though our sin reaches far, God's grace reaches infinitely farther.