It varies with the phone such as Samsung has KIES that may do that. Also...
https://www.google.com/search?q=save+text+messages+from+android+phone+to+PC%2Ftext+files&ie=utf-8&oe=utf-8
Shows it's one of those frequently asked questions. The bad news is that most replies are spammers.
Bob
I want to learn how to save text messages from android phone to PC/text files. Its kinda been frustrating figuring this one out.

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