"It is a faithful saying: For if we be dead with Him, we shall also live with Him: If we suffer, we shall also reign with Him: if we deny Him, He also will deny us: If we believe not, yet He abideth faithful: He cannot deny Himself." (2 Timothy 2:11, emphasis mine)
Wrap your mind around that.
I believe any Christian must at least stub their toe on this concept, if not fall headlong upon it. Jesus says something similar: "Whosoever shall fall upon that stone (referring, however symbolically, to Himself) shall be broken..." (Luke 20:17)
There seems, at times, to be an aspect to God that is impersonal, immobile, unkind even. So much so, that people even leave off believing in Him altogether, citing their misconceptions of His character as proof that He needn't exist.
The first passage was written by Paul to Timothy. As Paul was genetically Jewish and spiritually Christian, he had an insight into God's character that few people possessed. For him to make so bold a statment as "[God] cannot deny Himself", he must've both read about His character in the Jewish scriptures and coupled that with his personal knowledge of God—Himself—and the (then, new) saving grace and friendship of Jesus Christ. The point is, Jesus revealed a side of God that no one had ever seen before. And yet God the Father still remains the same Person He's always been. How can this be?
"He that hath seen me hath seen the Father." (John 14:9)
A common question of doubting unbelief goes something like this: "Can God create a rock that He can't pick up?" I'm sure, if you haven't heard it yourself, that you've at least formulated something along those lines. Omnipotence (all power) is a concept that can be hard to wrap one's mind around but is worth the mental effort in trying. Worth it, in spite of our own powerlessness.
So Paul makes his statement. Essentially saying, if there's one thing God can't do, it's deny Himself. And as sin is the active opposite to who He is, the one thing He cannot do is mingle His presence with sin. This is the main reason why He sent Jesus into this world—to be the one who lived the perfect life so that we wouldn't have to.
As an aside, I find it amusing that the implied premise to the question "Can God create a rock He can't pick up?" is that God is the Creator. It's a good start. Notice the following passage:
"God who at sundry times and in divers manners spake in time past unto the fathers by the prophets, Hath in these last days spoken unto us by His Son, whom He hath appointed heir of all things, by whom also He made the worlds;" (Hebrews 1:1-2)
God has spoken to us through Jesus. Not just the red-printed words in your Bible, but the active presence of Jesus, bodily, in the Middle East two-thousand plus years ago, was God speaking. I say all of that to say that one of the reasons that Jesus came, was to take the blame for the sin—that which God was unable to look upon or to touch. The one thing that He could not overlook in our lives, was essentially powerless to overcome on His own. As God holds up the world and the universe, He was unable to put that on hold to deal with the sin problem. So He sent Jesus.
The word Ebenezer (yes, as in Ebenezer Scrooge) is Hebrew for "stone of help". In the first book of Samuel (7:12) after a massive and miraculous victory over the Philistines, he "took a stone, and set it between Mizpeh and Shen, and called the name of it Ebenezer, saying, Hitherto hath the Lord helped us." This is symbolic of what Jesus has done for us. What God did for the Israelites under Samuel, Jesus actively does for us. And it's something that no one—not us, and not God the Father—can do. Give us, in our human form, salvation—help. But don't lose sight of God the Father. Jesus did what He did at the Father's love and request.
"Wherefore He is able also to save them to the uttermost that come unto God by Him, seeing He ever liveth to make intercession for them." (Hebrews 7:25)
See, God cannot deny Himself. And He can't deny us, either, when we come to Him—as it says in the above verse—through Jesus. And He's one Rock that He's not about to pick up. Jesus is here to stay.