What you describe is ‘clematis wilt’, a fungal disease. The disease spores winter over on the old vines but doesn’t affect the root system. Make sure you remove all the old vine from the trellis and put it in the garbage (not your compost bin or brush pile) to get it out of your yard. Start spraying every 10 days with Bonide’s Sulfur Plant Fungicide. Sulfur spray is a preventative fungicide so you need to use it before you see the symptoms. Other than when you’re spraying on the sulfur spray, you should never spray your clematis (or any garden plant for that matter) with water since wet leaves are the perfect place for fungal disease to take hold. If the disease shows up in spite of your efforts with the sulfur spray, cut the clematis right to the ground. This will force it to regrow from the roots which are immune to the disease. Continue to use the sulfur spray as a preventative as the clematis reestablishes itself. Large flowering clematis are more likely to catch the wilt than smaller flowered varieties.