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I know this is never explained at all in the movie but I always wondered how they first met. So here's my idea.

Pretty much Mel grew up as stray alongside his parents causing him to have rough childhood. His parents died saving him from a breaking ice river. This and his childhood pretty much caused him to be the paranoid dog we know. Later on Dipsy's owner would find him and temporarily adopt him until someone else would get Mel.

During his time at the General Store Mel would befriend the chubby pup that was Dipsy. She had self esteem issues since most pups would make fun of her weight. The two would develop feelings for each other that they kept secret until they would eventually confess to each other.

That's what I think would be a good backstory.