Nestorius (c. 386 - c. 451) struggled with the doctrine that Jesus is one man with two natures - divine and human. This is a hard problem - in what way could we say that "God is hungry" (Matt 4:2), or "God is tired" (John 4:6)? I'm not sure. God is spirit, a spirit does not feel these things. And God is omnipotent, these things seem contrary to that.
At the same time, Jesus is God, and:
"Wherefore in all things it behoved Him to be made like unto His brethren, that He might be a merciful and faithful high priest in things pertaining to God, to make reconciliation for the sins of the people. For in that He Himself hath suffered being tempted, He is able to succour them that are tempted." Hebrews 2:17-18Jesus, being God, was made like us and felt all the things we feel. You would be right to say "God suffered".
Nestorianism is the teaching that Jesus was (is?) two people - one God, one man. According to Wikipedia, Nestorius himself denied this ("Nestorius responded that he believed that Christ was indeed one person").
The main problem is that Jesus has two natures. This is the finding of Chalcedon, there is agreement there.
If we apply all of His human nature to God, then we make no distinction between God and Man. That is, if there are two natures - the two natures must be different in some way. If one is a subset of the other, then He really only had one nature. (struggling here...) If God is in every way like Man (through Jesus), then Jesus only had one nature - that of God (Man being a subset, lacking things like eternal preexistence and omnipotence).
To restate, if God is in every way like Man (through Jesus), and Man is therefore like God - only lacking in certain things (omnipotence, omniscience, etc.) - then Jesus did not have Man's nature. He had God's nature (which includes all the ways that Jesus is like Man) - because Jesus is not lacking in omnipotence, omniscience, etc.
I will have to read more on Chalcedon and exactly what Nestorius taught, and what was attributed to him. Of course, the attack I am making has come up before, being monophysitism. I will have to look into that as well.. (the beauty of Church history being that every argument has occurred before, it is just a matter of looking it up :)