True believers hang in there. Period. No exceptions.
As an example, Do YOU intend to stop believing?
Even true believers can stumble and fall, like Judas. As Paul reminds us in Romans 5:2, we rejoice in the "hope" (not the presumptuous certainty) of sharing the glory of God. If salvation is absolutely assured after accepting Jesus as Savior, why would Paul hope?
There's a couple problems with that statement.
1. It suffers from the "no true Scotsman" logical fallacy. ("No true Scotsman puts sugar on his oats." "Argyle puts sugar on his oats." "Then Argyle is not a true Scotsman."). That's a category error.
2. Even if we ignore the above logical fallacy, that doesn't help any individual. No true believer will stop believing. Check. Doesn't mean I'll always continue to believe - I might not be a "true believer."
3. My intent is irrelevant. Sometimes we trip and compromise our intents.
Here's the simple problem: I don't intend to ever stop believing, but I know that left to my own devices, I still could because I've come awfully close in the past. I've felt the metaphorical hand of God pulling me back from unbelief and rebellion.
At the end of the day, as a Calvinist, I believe that God must give us his grace if we are to endure to the end. Still, I think the Catholics are right to point out that it would be presumptuous to assume any of us are automatically going to make it.