What kind of guitar is Chet playing here?
 

It looks like an Acoustasonic prototype of some sort. Also, it looks as though there are lines on the fretboard, but they're flat. Has an odd, synth-like sound to it.
<iframe width="560" height="315" src="https://www.youtube.com/embed/OS_mxt5Z0ok?start=214" frameborder="0" allow="accelerometer; autoplay; encrypted-media; gyroscope; picture-in-picture" allowfullscreen></iframe>
https://youtu.be/OS_mxt5Z0ok?t=214

P.S. I need to figure out how to embed a video. :(

That's a Gibson Chet Atkins.

Chambered solidbody design. They made them in steel and nylon string models. They're supposed to be very comfortable for electric players.

Thanks, Willie, that was a quick answer! Is it fretless?

Nevermind, I just looked it up on Reverb. Seems they have frets. The one on the video looks like the frets are just painted on-- must just look that way to me due to the age of the video and of my eyes.:)

My neighbor has a few of them.

http://mikeperrymedia.com/photos/dave_matthews01.jpg

(I've never met Dave, but I have seen him at the high school our daughters attend together.)

Hmm... looks very much like a Seattle photo-- seems his guitar got rained on. :) BTW, unrelated, but I see you are from Greenlake. I used to live in Wallingford. I love that area. Lucky you!

Yeah, but I'm a Methodist pastor so I only get to live here as long as the Bishop lets me.

Love me some DMB. I’ve been a fan for many years. I’ve owned a few SST’s (inspired by Dave). They're a guitar all their own.

It looks like Gibson did make a fretless model for Chet.
https://www.flickr.com/photos/133334...4/20992387631/

Epiphone has reintroduced their version of the Chet type chambered body nylon and steel string models. I keep looking for a demo video of these.

Aha! So, my eyes aren't as bad as I thought. Also, his pitch sounds a little bendy in a few spots on that video, with his fingers not going up/down, but left/right, so that would explain it. Thanks for tracking that down, gmr!

THEY ARE NOT THE SAME GUITARS!!! The SST model (like what Dave Matthews is playing) is a totally different beast.

This is what Chet was playing, the Gibson Chet Atkins Country Gentleman model, and no not fretless...

https://i.imgur.com/4W3xUvQ.jpg

EDIT: sorry when you posted the video the picture that popped up for me was his Gibson country gent model, now I see you were looking at the first guitar in the video.