I Have this problem too and my do not disturb is not turned on. I tried changing all sorts of settings Including have AT&T deactivate voice over lte on my account. iPhone 6 on AT&T. Oddly enough it only happens while I am at work. It did not happen when I had my iPhone 4s though. Strange. I tested it over 90 minutes yesterday by continually calling my phone from work. I'd say 80-90% of the time I would get two rings than voicemail. My cell never rang and never showed a missed call. A voicemail notification showed Up instead.
AT&T called me today to check on the situation. Call came through. He tested two more times. The last call did not come through. I noticed the first two times, my phone was not in its usual place. So I ran a test myself. I had my phone in its usual spot. Called three times with the phone in sleep mode. Only the third call actually came thru and not to voicemail. I tried a 4th time with the phone not in sleep mode and it still rang twice and went to voicemail I moved my phone to the other side of my desk and tried again. First three calls all came through. I was so excited so I tried a 4th. That one went to voicemail. I then tried with the phone not in sleep, call came through. For fun I tried one more time in sleep mode, call went to voicemail. So results: left side of desk, 3 out of 4 calls went to voicemail. On the right side, 2 out of 6 went to voicemail.
SO if you add that all up, it worked 50% of the time, but odds were better just llike 3 feet over more. Weird.
sorry this came thru in two different fonts. I copied and pasted from an email I sent. Lol.