The way you describe it, yes. Going through a hard time and living badly? Sure. But if they were out there doing evil and just ignoring God, then most Christians would say such a person is not saved.
There are three major views on this: exclusivism, inclusivism and pluralism (or universalism).
The exclusivist position says you are correct. Such a person will not go to heaven when they die. Knowledge of and acceptance of Jesus is necessary for salvation.
The inclusivist position essentially says that although Christ is the sole means of salvation that does not mean that knowledge of Christ is necessary for salvation. In other words, the person you described may, in fact, be a Christian despite the fact they don't even realize that it's been Christ leading them all along. Here's how Billy Graham put it:
I think everybody that loves Christ, or knows Christ, whether they're conscious of it or not, they're members of the Body of Christ. And I don't think that we're going to see a great sweeping revival, that will turn the whole world to Christ at any time. I think James answered that, the Apostle James in the first council in Jerusalem, when he said that God's purpose for this age is to call out a people for His name. And that's what God is doing today, He's calling people out of the world for His name, whether they come from the Muslim world, or the Buddhist world, or the Christian world or the non-believing world, they are members of the Body of Christ because they've been called by God. They may not even know the name of Jesus but they know in their hearts that they need something that they don't have, and they turn to the only light that they have, and I think that they are saved, and that they're going to be with us in heaven.
Pluralist/Universalist positions tend to be non-Christian positions, so I won't really dive into those but essentially they say that all spiritual paths will lead you to the same place.
There is biblical support for the first two positions (exclusivist and inclusivist). Neither side has a slam dunk case although you will find people on both sides who proclaim that theirs is the one true way of understanding this. I don't think there is any biblical support for pluralist/universality positions.
