There are people who will tell you that Continental Trailways didn't go out of business... it just got smaller.
It actually started out in 1936 as the National Trailways Bus System, a loose knit group of regional bus companies that could get people from one end of the country to the other. In time there were the local companies and the national system. They were the worst competition Greyhound had.
In 1987, the company (Continental Trailways, not the other Trailways companies) were taken over by Greyhound.
The thing I remember that they didn't go on the same routes. They went to Cookeville, Tennessee (Greyhound didn't, until 1987!) They had very unique buses. And the drivers drove a lot faster!