From the 1950's through the 1970's, country music was referred to as country-western or country and western. You still hear that moniker today, but not as frequently. You mainly hear it from non-country listeners who reference the genre.
There's good reason why western is no longer associated with country. For starters, the city that produces most country music is EAST of the Mississippi River. And few songs have Western themes anymore.
I recently purchased "Gunfighter Ballads and Trail Songs", the classic Marty Robbins album, and I think it epitomizes Western music. Most of the songs are original, but there are some traditional ones like "The Strawberry Roan" and "Billy The Kid".
One reason for the lack of Western music is because few of today's artists understand the lifestyle of the old American West. According to his biography, Robbins' grandfather was a Texas Ranger, and that was partly responsible for motivating him to write western songs.
In the 1990's, there was a return to Western music. It was mostly in theme and dress, not in sound. Many artists replicated the look of real cowboy/rancher George Strait, and Wranglers and Stetsons became a part of country music dress.
Look at today's artists and few of them have that full cowboy look of the 1990's. Sure, they still sport the hats, but it's often accompanied by apparel that's more mainstream.
Garth Brooks sang several songs about rodeo, especially early in his career, and his mention of Chris LeDoux in "Much Too Young (Too Feel This Damn Old)" broadened the former rodeo champion's fan base.
The one that was most-similar to Robbins' stuff was "Cowboy Bill", a ballad about a Texas Ranger, turned storyteller, that was on Brooks' debut album in 1989.
But as Brooks drifted towards country-pop, so did the themes of his songs. It's only on his most recent album, The Lost Sessions, that he returned to songs with rodeo/western themes - "Good Ride Cowboy", a tribute to LeDoux, "That Girl Is A Cowboy" and "Cowgirl's Saddle".
Will the western rejoin country? I don't think it's likely, but neither was the re-emergence of western themes during the 1990's.