The issue occurred during a 4 hour call:

I have AT&T
I live in  marginal coverage area, not unusual to see "Emergency Calls Only" on the phone in the house
The call was to a FedEx Express conference code number
It was made on a Thursday afternoon while sitting on my deck
I share two fences with a golf course, it is possible the network in the area had additional demand that afternoon.

I have attempted to duplicate the issue three times and have been unable to do so:

I initiated my conference calling number at work.  Sent three texts, no disconnects.
I called my house landline this morning in the kitchen, sent 3 texts with no disconnects.
I initiated my conference calling number this morning, went onto the deck in the same location I had the problem, sent three texts, and had no disconnects.

Hi Terri...

This may be an Android thing - I've seen the same behaviour on my Xperia Z3 running 5.0.2, specifically when I'm on a phone call and another one comes in, where I try to reject the second call with an SMS: everything gets disconnected (but the SMS does go through).

Would love to understand it more, especially if its going to start affecting us in a work sense.
-JP