I call "High Country" anyplace that is at or above timberline. My favorite place to be!
Here in the Oregon Cascades, timberline is close to 6,000 feet. In the North Cascades of Washington it's a bit lower, nearer 5,000 feet. In the northern Colorado Rockies it's about 11,000 feet. In Wyoming's Wind Rivers it's about 10,500 feet. In the Rockies there's a low level timberline, which is the altitude below which there is not enough precipitation to support forest. Below that hat it's open country but not high country! We don't have that problem out here on the wet coast!
Of course the exposure of a given location has a lot to do with just where timberline is, which is why I stress the "about."