Yes, but I can't think of any place where you would have to say that, or even should say that.
You're more likely to hear it when there is some negative emotion towards the object, as in "That dog of mine has dug up the garden again!"
With "the", I can't make an argument for it colloquially. You might hear something like "Do you remember that book of mine - you know, the one with the purple, sparkly cover - the one that I've read six times? Well, I've lost it! I'm really sad."
When a student asked me, "Can I replace 'my book' with 'the book of mine'?", I did not think of any situations to use "the" in, either. Since we are non-native speakers of English and our language does not have the concept of "article" , we are always confused about articles.