Chris: I appreciate your help! I should also have pointed out that the problem seems to be an iMessage issue rather than what I described simply as 'texting'. Case in point: my grown son and I both have iPhone 6's. I tried to send him an imessage and got repeated 'Not Delivered' messages (and indeed he did not receive the messages). I've been able to press the red question mark icon on the 'Not Delivered' message and choose 'Send as Text' and this seems to get the message through. Nothing explains why iMessage will suddenly no longer be able to get through to another iPhone that I know to be on and connected. By the way, per your advice, I did turn off iMessage, reboot my phone, and tried again to send an iMessage. No help.
A colleague of mine reports the same issue with her iPhone 5s.