Is it possible to overcome all temptations that we are faced with?

YES!!!!!

How is that possible?

We simply "choose" not to yield to the temptation.

This is a power that is available to us all and was given to us by our Creator.

The reason anyone "succumbs" to temptation is due entirely to their "choosing" to do so.

"If the grace of Christ cannot remedy these defects, what then constitutes transformation of character?" That I May Know Him, p. 157

Tom, I'm not sure what you believe about sins of ignorance. I believe unintentional sins and sins of ignorance are two entirely separate types or categories of sin. In the OT, there were two different sacrifices for dealing with them. An unintentional sin is a known sin we commit spontaneously.

Bob, I am more inclined to believe that the only way to resist temptation is to choose to stay connected to Jesus, rather than choosing not to sin or choosing to be righteous instead. It is by beholding Christ and Him crucified that we are empowered to resist any and all temptations. Sin loses all its power and appeal at the Cross!

By unintentional you mean unpremeditated, correct? I believe sins of ignorance need to be made known in order to be overcome.

Yes, unpremeditated. Yes, sins of ignorance are ongoing sins committed without realizing it is a sin. For example, working on the sabbath, eating pork, believing we go immediately to heaven or hell the moment we die, etc. It is impossible, however, to be impatient, unkind, unlovely, or to lie, cheat and steal and not know it is a sin. The reason being is the Holy Spirit dwells within us, seated upon the throne of our soul temple, endowing us with spiritual discernment so that we cannot commit such sins and not know it is a sin.

Romans
8:9 But ye are not in the flesh, but in the Spirit, if so be that the Spirit of God dwell in you. Now if any man have not the Spirit of Christ, he is none of his.
8:16 The Spirit itself beareth witness with our spirit, that we are the children of God:

More tactful, yes, but unknowingly unkind? cheat? steal? lie? When can we expect God to set us free? If sin is so subtle, how can we know if we are free?

True. But do we experience the miracle of rebirth before our "old man" habits of sin are confessed and crucified? And, how long does it take us, cooperating with God, to see and confess all of our sinfulness, in light of the cross? According to Steps to Christ:

SC 28, 29
You who in heart long for something better than this world can give, recognize this longing as the voice of God to your soul. Ask Him to give you repentance, to reveal Christ to you in His infinite love, in His perfect purity. In the Saviour's life the principles of God's law--love to God and man--were perfectly exemplified. Benevolence, unselfish love, was the life of His soul. It is as we behold Him, as the light from our Saviour falls upon us, that we see the sinfulness of our own hearts. {SC 28.2}

We may have flattered ourselves, as did Nicodemus, that our life has been upright, that our moral character is correct, and think that we need not humble the heart before God, like the common sinner: but when the light from Christ shines into our souls, we shall see how impure we are; we shall discern the selfishness of motive, the enmity against God, that has defiled every act of life. Then we shall know that our own righteousness is indeed as filthy rags, and that the blood of Christ alone can cleanse us from the defilement of sin, and renew our hearts in His own likeness. {SC 28.3}

One ray of the glory of God, one gleam of the purity of Christ, penetrating the soul, makes every spot of defilement painfully distinct, and lays bare the deformity and defects of the human character. It makes apparent the unhallowed desires, the infidelity of the heart, the impurity of the lips. The sinner's acts of disloyalty in making void the law of God, are exposed to his sight, and his spirit is stricken and afflicted under the searching influence of the Spirit of God. He loathes himself as he views the pure, spotless character of Christ. {SC 29.1}

Evangelism, page 272
"A Work of the Heart.--There are many who try to correct the life of others by attacking what they consider are wrong habits. They go to those whom they think are in error, and point out their defects. They say, "You don't dress as you should." They try to pick off the ornaments, or whatever seems offensive, but they do not seek to fasten the mind to the truth. Those who seek to correct others should present the attractions of Jesus. They should talk of His love and compassion, present His example and sacrifice, reveal His Spirit, and they need not touch the subject of dress at all. There is no need to make the dress question the main point of your religion. There is something richer to speak of. Talk of Christ, and when the heart is converted, everything that is out of harmony with the Word of God will drop off. It is only labor in vain to pick leaves off a living tree. The leaves will reappear. The ax must be laid at the root of the tree, and then the leaves will fall off, never to return."