Because the dog was never mine.

This sentence has 1 clause as follows with subject and verb marked.

Because the dog was never mine.

As a rule, a sentence must contain at least one independent clause and hence it must be able to communicate a COMPLETE idea. Now keeping in mind, this sentence is a fragment because it does not communicate complete idea. After you read the sentence, you dont get the complete picture of what the author is trying to say. All you know is that "the dog was never mind" is the cause of something. But you dont know what that something is.

Contrast with the following sentence:
The dog was never mine.
Now this sentence communicates a COMPLETE Idea and hence is not fragment.

Thus, as you can see, by adding the word "because" you converted the 2nd sentence into 1st sentence and this first sentence leaves the user 'wanting' for more information.

I hope this helps.

Regards,

Payal Tandon