I have often heard people claim that Jesus' parable of the sower proves that some can become unsaved after they first become believers. However, that is not the case. There are a few important things that we need to see from the parable.
If you'd like to read it for yourself you can find it in Matthew 13, Luke 8, and Mark 4. Pay special attention to the version found in Luke 8.
If you read only the Mark and Matthew version you may miss an important aspect.
The parable explains 4 groups of seeds sown by a farmer. (1) Seed falls on the road. (2) Seed falls among rocks. (3) Seed fall among thorns. (4) Seed falls on good soil.
Notice what Jesus says about the first group in the version Luke gives us. He includes a very important line.
"Those by the wayside are the ones who hear; then the devil comes and takes away the word out of their hearts, lest they should believe and be saved."
So this is talking about the first group, (the seed that falls by the wayside). Notice that it's the only group of whom it says they are unsaved. They could have believed and become saved, but they missed out.
It's also valuable to note that they didn't believe at first and then stop. Instead, they never believed in the first place. They heard the message of Jesus but never believed. They were never convinced that it was true. So group (1) is the only group that this is said of.
Notice that the other three groups, (2) seed among rocks, (3) seeds among thorns, (4) seeds in good soil are all lumped together. Of these three groups, Jesus never calls them unsaved. If you don't believe me, read the parables again. Point to the verse that says these three groups are unsaved... you won't find it.
In fact, He shows that each has life. It's true that the life gets choked and withered in two out of three of the saved groups. None the less, he never says they are unsaved.
What's more, He doesn't say that any of these three groups die.
In fact, group 2 is still alive but withered at the end of the story, and group 3 is alive but fruitless at the end of the story. Don't miss this, Jesus does not say that group (2) or (3) die. Most people simply assume that they die, but it's not what the story says. Don't believe me? go back and read it again.
Group (2), (3), and (4) has life, even at the end of the story. Each retains life, but two out of the three have a choked and withered life.
Each was made alive by Christ, even if their mortal lives are less than obedient.
Fruit is in view. He's not telling group (2) (3) or (4) that they are unsaved. He's telling them that they are in danger of going through life without bearing fruit. This would be a tremendously sad outcome.
So, there are consequences for not bearing fruit as a Christian. It just so happens that the consequences are not a loss of salvation.
Here's the full graphic to show how it all fits together.