Do we know of any tidally-locked planets with atmospheres? If not, is there any reason such a planet couldn't exist? 

I ask only because that planet would literally have a twilight zone and I want to know that that's a thing somewhere.
As we haven't measured the atmosphere of many rocky planets yet, there aren't many examples. But I found 55 Cancri e:
It is a super-earth with very close tidally locked orbit and an atmosphere that has been measured by Hubble. A year on the exoplanet lasts for only 18 hours and temperatures on the surface are thought to reach around 2000 degrees Celsius. The detected atmosphere seems to consists mainly of hydrogen and helium. So you do have a constant Twilight Zone, just a rather hot one. 



Sources: 
- Beeing tidally locked
- Detection of the Atmosphere