The claim is often that Jesus did what we couldn't do -- he perfectly followed the Law in our place, as that is what God demands.
However, if the claim is also that Jesus is God, then does saying he fulfilled the Law hold any validity?
This isn't a matter of if Jesus was perfect, he lacked the ability to break the Law. It's a matter of if Jesus was God, then some of the Laws had a hard time applying to him in the first place.
For instance, take the commandment "Thou shalt not steal." If God is in fact the Creator of all, and He's made everything, and anything you own is in fact provided to you by God, then God "owns" everything. How, therefore, could Jesus even begin to go about stealing, since it was all his to begin with?
Same with not being allowed to covet -- we again go back to the idea that it's all God's by default. He made it, He owns it, He has the rightful claim to everything. If you own everything, how can you covet something your neighbor has? It's already yours.
"You shall not murder." I'm not going into a debate on some of the acts committing by God in the Tanakh, but the idea is often that if God does kill, it's not murder, it's something He's allowed to do, the same way a painter is allowed to destroy a painting. If God killing is in a completely different category, and His right since He is just and righteous, then where does God even begin to have the opportunity to break that commandment?
If Jesus is God, can we still say he did what we couldn't do? How do you perfectly follow something that doesn't even apply to you? God wouldn't have a chance to even try and break the commandments, because if there's something out there that God can steal, then you lose the very definition of 'God' in a Christian sense. I'm sure the duality of Jesus would come into play here, with the man aspect of Jesus actually under this restriction, but there's no way to make that make sense. You'd have to fall back on the "It's a mystery" idea.