Long story short: no. Most native speakers of English who have no experience in any foreign language do not get much out of overhearing conversations in other languages.
David Ives, via the play "Universal Language," (an artificial koiné for theatrical entertainment) might argue with me on this one, but with English being the bastard child of a language that it is, most native English speakers who have no experience with any foreign languages are not able to pick up more than a vague meaning when hearing unfamiliar languages, even those within the same "language family."
You would think, for example, that conversational German would be somewhat (can't say easily) accessible to a native English speaker, but aside from conversational pleasantries afforded us all by UG, most anglophones do not understand German. Despite the heavy French influence, most anglophones do not understand French, either, unless they study it or have some other intense exposure.
What you've described is an excellent question from a linguistics standpoint. For example, I speak French, and I can understand the other Romance languages that I have not studied. It is interesting to note that native Spanish speakers often have trouble understanding conversational Portuguese, but native Portuguese speakers can work with conversational Spanish.
It does depend on your definition of "language" in relation to dialects. There are many dialects that most native English speakers can understand, and there are also quite a few that are unintelligible depending on the region. Tangier Island is a great example of a remarkable preservation of the English of the Colonies.