I had a similar experience with a car several years ago. It bewildered me completely, but it was consistent after a certain speed. Repair shop found nothing wrong with tire balance or wheel alignmnets.
After several frustrating trips to the repair shop and several attempted fixed failures, they finally checked the actual tread of the tires. Turns out the actual tread (not cross wise, but long wise) had dips and high spots on them at different spots on the tires. When the tires got in sync with each other, just right, the uneven tread lines (long ways around the tire) would cause the car to shake violently.
They took each tire and make each tire true round and got the tread level and sheared off those high spots. After that, it rode just fine. They fixed the tires at no cost.