Jesus is fully God and fully man; He is God-man.1
As to His humanity, just as a baby must grow in wisdom and understanding, the same was true of Christ Jesus. In becoming a man, He voluntarily limited the independent use of His Godly attributes while upon the earth; God the Father gave Him use as He determined.
Since His ascension to heaven and His exultation, Christ Jesus' Godly attributes are no longer limited. He knows all things, is all powerful, and is present everywhere in His Spirit.
As to whether, or not, Jesus could have sinned, some believe that He could have, while others believe that He could not have sinned. Some believe that if He could not sin, then there was no purpose in His temptation by Satan. Others, however, say that the purpose was to show that He could not sin.
In His totality as God-man, Jesus could be tested, however He could not sin, for God cannot be tempted by evil.2 Within Christ Jesus dwells perfect humanity (a sinless human nature) and Divinity. Two natures, one will.
1 (Philippians 2:6-8)
2 (James 1:13)