From the 1689 London Baptist Confession:
Chapter 17: Of The Perseverance of the Saints
1. Those whom God hath accepted in the beloved, effectually called and sanctified by his Spirit, and given the precious faith of his elect unto, can neither totally nor finally fall from the state of grace, but shall certainly persevere therein to the end, and be eternally saved, seeing the gifts and callings of God are without repentance, whence he still begets and nourisheth in them faith, repentance, love, joy, hope, and all the graces of the Spirit unto immortality; and though many storms and floods arise and beat against them, yet they shall never be able to take them off that foundation and rock which by faith they are fastened upon; notwithstanding, through unbelief and the temptations of Satan, the sensible sight of the light and love of God may for a time be clouded and obscured from them, yet he is still the same, and they shall be sure to be kept by the power of God unto salvation, where they shall enjoy their purchased possession, they being engraven upon the palm of his hands, and their names having been written in the book of life from all eternity.
(John 10:28, 29; Philippians 1:6; 2 Timothy 2:19; 1 John 2:19; Psalms 89:31, 32; 1 Corinthians 11:32; Malachi 3:6)
2. This perseverance of the saints depends not upon their own free will, but upon the immutability of the decree of election, flowing from the free and unchangeable love of God the Father, upon the efficacy of the merit and intercession of Jesus Christ and union with him, the oath of God, the abiding of his Spirit, and the seed of God within them, and the nature of the covenant of grace; from all which ariseth also the certainty and infallibility thereof.
(Romans 8:30 Romans 9:11, 16; Romans 5:9, 10; John 14:19; Hebrews 6:17, 18; 1 John 3:9; Jeremiah 32:40)
3. And though they may, through the temptation of Satan and of the world, the prevalency of corruption remaining in them, and the neglect of means of their preservation, fall into grievous sins, and for a time continue therein, whereby they incur God's displeasure and grieve his Holy Spirit, come to have their graces and comforts impaired, have their hearts hardened, and their consciences wounded, hurt and scandalize others, and bring temporal judgments upon themselves, yet shall they renew their repentance and be preserved through faith in Christ Jesus to the end.
(Matthew 26:70, 72, 74; Isaiah 64:5, 9; Ephesians 4:30; Psalms 51:10, 12; Psalms 32:3, 4; 2 Samuel 12:14; Luke 22:32, 61, 62)
This is what I understand to be the classical Calvinist position on perseverance. The third item says that the elect can commit grievous sins but will eventually repent. This is also in keeping with the doctrine of depravity ... that every part of our nature has been corrupted by sin. Although God's grace in his effectual calling brings us to repentance and faith, which we could never have prior to regeneration, He does not remove sin's corruption from us in this world.