Even what might be considered different languages today that originated from a common tongue can sometimes be understood without someone knowing both "languages." I knew a woman from Sweden when I was getting my MA. She had never taken any German classes, but she could still get the gist of someone speaking Dutch and German because they were related to her native language. She did not understand everything, but she could tell the languages were related. When I was getting my BA, I looked at a friend's basic French textbook and was able to read most of it even though I had never taken any French classes, but I had several years of Spanish.
Southerners can still understand Yankees while speaking. An Englander can still understand a Scot when he is speaking English (By Englander I mean someone from England specifically, not a citizen of the UK). Those are the same language, but accents can be remarkably different.
I would be willing to go along if someone said they could be distant derivatives of the same language or if they shared some common words or origins like English and German and Latin. The mushing together would stop someone from being able to recognize the language, but then it would lose the right to be called the same language. Possibly an example like Old, Middle, and Modern English as well, but there is a reason the languages are separated.