You know what.... top question! I'm gonna have to have a look at that.

You are right, there is a probability associated to the tunnelling effect, so if it's harder to tunnel then probability reduces so the current should therefore be reduced. Of course there is a limit to the thickness of the link, so at some point the probability of tunnelling must tend to zero, which stops any current flowing.

I'll see if i can give you a better answer later. Awesome question though :)