What if connection 3, loyalty 5?
At C:3 L:5, you're the old friend that they'll drop anything for, no question's asked, and help out however they can. Abuse it, however, and you'll start seeing their Loyalty drop pretty fast.
The degree of success also helps determine just how glad they are to see you and how much help they'll offer without demanding any Quid Pro Quo.
With so many points in such a contact, it's best to always have them available.
Oh, you'll always be able to get hold of them. The question is, are they willing and able to do what you need from them. In the case of C:6 L:5, if you manged to roll a 1 you might be hitting the contact up during a hostile takeover attempt, or in the middle of a massive labor dispute, or some other problem that's got them pulling their hair. Essentially, they've got too much on their plate to pull any strings for you.
Now, the PCs being professional "problem solvers" and all, the players have the option of doing a little side job to smooth his road so he can smooth yours.
For a tie, like this, it'd be fairly smooth sailing, but rolling a 1 on a C:6 L:1 contact... well, at that point you're probably just going to want to walk away. Clearing things up will probably take more work than the job you need help on.