5pinkarrows, What you use to fence your coop/run has to be a trade-off between your potential risk, the price of the materials and the risk you are willing to accept. For some people, it may balance out in favor of chicken wire. You just have to keep in mind that you've increased your potential risk with that choice of fencing. That doesn't mean you'll necessarily always lose your animals with such a tactic--your risk is just higher. I think that for a lot of applications, 2" X 4" galvanized welded wire is a good option for the price. It'll perform reasonably well against most things. I have not yet seen photos of anything tearing this kind of wire here on byc (although I may have missed it--it doesn't seem common at all). You still have the issues of grab-through, but that can be dealt with smaller opening wire strategically placed. Maybe 2X4 welded wire with chicken wire over top is a good option for some. Hardware cloth will likely always be the option with the lowest potential risk, but it is pricey. I should note that I always drool over the 1/2" X 2" 12 gauge galvanized welded wire that I see at O.K. Feeds, but the price will take your breath away. Keep your eyes peeled at garage sales, estate sales and CL for people getting rid of their wire. The great thing about wire is it doesn't have to be new, used works pretty well too. Home Depot and Lowes also sell it. I'm not sure if your run is covered, but keep in mind that a 6' fence wouldn't even slow down a coyote or bobcat from leaping over it. My guess is that bobcats are pretty smart.
Another thing to consider. You can have the most predator proof coop possible, but it all won't mean a thing if you free-range and if you forget to lock them in at night. I have to keep reminding myself of that everyday. I'm sure that is how I'll suffer my first loss. But like a lot of us here, it seems to be a risk I'm willing to take. They just love to wander around the yard.