In the case of berry vine, you'd need to dig all around the smother area, to make sure that nothing under the smother is connected to something outside the smother area. I'd suspect that's why your previous attempt with cardboard failed, assuming it was laid well. In the case of berry vines, I'd go for an overlap of about a foot. It needs to have enough weight put on top to completely conform to the soil surface, so nothing can get going, creeping around under there to find an edge.

IMO/E, cardboard is superior because there's no further work of removal involved. Strong, corrugated cardboard, not something thin like a cereal box. A smother that needs to be removed re-exposes buried seeds to a better chance of finally being able to sprout. Some seeds can remain in the soil for decades, just waiting for conditions to be just right. Any time the ground is disturbed, something is likely to sprout.

Smothering works, except when it fails for one of several reasons, which you'd want to avoid:
Woody stumps poking holes.
Seams not overlapped enough.
Disturbed too soon.
Not covered with enough mulch/leaves/other organic material to add enough weight & completely block the light.
Thin layers of newspaper used for plants that are strong enough to come through. I always use corrugated cardboard.

Later maintenance fails that can seem like smother fails:
Lack of borders.
Mowing seeds back into beds.
Failing to pull unwanted plants, (usually via bird-dropped seeds,) and letting them drop new seeds.

In regard to berry vines, a surface border like line of bricks or landscape timber might not be enough if adjacent to neighbors' yard or wild area from which they can creep back. You'd need something that's also at least 6" deep, as well as several inches high, to prevent invasion from both above & blow ground.

see this for ideas: Wild blackberries