We rent 10 acres from a little old lady. We've been there for 7 years, and in all those 7 years I've never had a horse run through the fence, get tangled in the fence, or be badly hurt by the fence. My mare is clumsy and will come in with a nick every now and then, but my pony is meticulous about what comes near him be it fence, mud, or me.
Our horses respect the fence, so I see no problem with it. We're not going to replace the whole fence only for her to sell the property or her pass away and her family sell the property.