Angus,
These vines are VERy sensitive to salt.
The first sign is the leaves turn brown from the ends up.
Your water from your home, fertilizer for either the vine or the grass, or even salt you use in the winter accidentally reaching the vine somehow, can kill the vine.
Do you have a water softener system?
Other than salt, there is no reason for the vine not to climb for you.
Try a different kind of trellis, it is a woody vine, mabey it doesn't like lattice.

Thank you for the info, butterfly. We do have a water softener but water to outside spigots isn't softened. Fertilizing didn't harm them: new growth occurred throughout growing season. The only time they died back was quickly after attempting to train them onto the trellis. The trellis is natural wood- cedar. Wonder if that could be the issue? Cedar wood does have some 'different' compounds in it.