This happened to us a couple of times over the last few months and we eventually discovered that on each occasion my mother had mistakenly sent a text to our land line. We have a DECT phone which does have some SMS capability but I have never tried to use it and have no idea how to set it up. I don't see how it can have anything to do with caller ID settings but it may have something to do with SMS settings on a DECT home phone.
This happens in my mother's house too. She can send texts from her landline no problem, but if someone replies, the text doesn't come through and the phone rings about every hour with no one at the other end. She has to disconnect it from the wall.