While we are able to hold onto the promises of God in assurance that we are among the elect, I don't know if any living human being can actually 'realize' that they are not among God's chosen people. One can most definitely realize that they are not living in Christ, but that they will not ultimately be gathered among his sheep is hidden in the secret counsel of God.
On the other hand, I suppose that the Devil himself understands God's sovereignty quite similar to Calvin, yet he surely knows that there is no hope for his evil soul.
The bottom line is that it is Christ alone who saves, not the title of Calvinist or otherwise.
