Arminians are sometimes characterized as believing that a true Christian can commit apostasy and fall irreparably away from grace and salvation. This, however, is not an accurate depiction of Arminian theology. There certainly are Arminians who hold the view that a true believer can commit real apostasy, but there are are also many who identify themselves as Arminians and believe that God will guard and keep all true Christians enabling them to persevere to the end. In the latter view, perseverance is a demonstration of authentic saving faith. Both views are consistent with other affirmations that all Arminians hold in common including total or radical depravity, conditional election, unlimited atonement, and resistible grace.
It may help to know that the discussion over perseverance within Arminianism goes all the way back to the Reformation and can be seen in one of the earliest and foundational documents in the Arminian tradition. The Five Articles of Remonstrance, written in 1610 by followers of James Arminius, outline the Arminian opposition to unconditional predestination. In describing the Arminian belief in the grace given power of the Christian to successfully strive against temptation, the fifth article states,
"But whether [those who are incorporated into Christ by true faith] are capable, through negligence, of forsaking again the first beginning of their life in Christ, of again returning to this present evil world, of turning away from the holy doctrine which was delivered to them, of losing a good conscience, of becoming devoid of grace, that must be more particularly determined out of the Holy Scripture, before we ourselves can teach it with the full persuasion of our mind" (emphasis mine).
This statement is important for two reasons. First, it demonstrates the Arminian commitment to biblically grounded doctrine. To depart from biblical authority is to depart from authentic Arminianism. Second, it demonstrates that Arminians have historically acknowledged the tension in Scripture over the issue of perseverance and have made room in their ranks for both positions. The Remonstrants were not primarily interested in opposing the budding Calvinism of the Reformation. They were first and foremost interested in rightly understanding and faithfully teaching Christian Scripture. This commitment to the Bible yielded a tension within the Arminian tradition which reflects a tension in Scripture, and, thus, there is room within Arminiasm for those who believe in the final perseverance of the saints and those who believe in the possibility that a true believer may commit real apostasy.
Critics of Arminianism should take note of this distinction in their characterizations of Arminian theology. Fairness in debate requires good will and a faithful attempt to properly portray the position of those who disagree. Describing Arminian theology narrowly as believing that one can fall from grace is a misrepresentation of the Arminian tradition.
I suspect that there are some (perhaps many) who resist the Arminian label because they believe in the final perseverance of the saints and have heard Arminianism defined narrowly as including a belief in real apostasy. It is important for Arminians to be clear that there is room in our camp for both positions. On occasion, I will hear someone referred to as a "two point Calvinist" indicating that they believe in total depravity and perseverance of the saints. In response, it is important to note that a two point Calvinist isn't much of one. In fact, a two point Calvinist makes for a fine Arminian.