bustermcleod Said:
"How do I download a backup of my Messages?: I'd like to download the message history from the Messages app, and am not sure how to do this. Searching the internet, I see 3rd party backup software that claims to do this, but I'm wondering if there's an Apple-approved way to do this."
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As for Apple...
Are these on a Mac?:
Boot up a Mac associated to this account, making sure it does not connect to the internet upon boot (perhaps powering off your router?). While WiFi is off, try exporting your messages. to a different location (i.e. external hard drive).