I moved to Germany in 1991 (for a year that turned into a lifetime, but that's another story, my life story, as it were) and to Switzerland in 1995. After four years in Germany, and with a German girlfriend (now wife) whose family speaks little or no English, I could speak High German pretty well.
But the Swiss don't speak the same language; they speak Swiss German (in numerous dialects). There is a Swiss High German, and German speakers in Switzerland do speak it, but it is not their everyday language. They code switch rather easily between their varieties of German (not to mention French, Italian, and English). Many Swiss German speakers switch to High German or English when speaking to me, and even after 18+ years here, I don't speak Swiss German (or Basel German), which I would need to sit down and actually study if I were ever going to do so.
In contrast, when my wife is in Basel, after 13+ years here, she now speaks a mix of High German and Basel German, and she understands the local dialect much better than I do. In this sense, you can see that the two varieties of German, while quite different in many ways, are close enough for a native speaker of High German to pick up enough Basel German to get by, while a non-native speaker like myself can understand it, but picks up very little without extra effort. (That's an anecdote, of course, and not an experiment, so consider that a hypothesis to be tested, rather than a claim being made.)
All this came to mind on reading Geoffrey Pullum's "Undivided by a Common Language." Pullum vigorously argues that the differences between British English and American English, in their standard varieties, "is wildly, insanely overstated." I agree—and especially on the basis of the contrast with the differences between High German and Swiss German. There are many differences in grammar and comprehensibility between the two Germans (and this doesn't even bring up dialects in Germany and Austrian German); there are, as Pullum argues, hardly any grammatical differences between British English and American English, and the ones that do exist are insignificant in terms of comprehensiblity. The differences in vocabulary are considered significant enough for the American publisher of Harry Potter to translate some terms into American English, but even those differences are minor (and almost everyone I mention that to says, "That's ridiculous. People know how to use dictionaries."). But it's a matter of a few words here and there, while a novel written in Swiss German has to be translated into High German for anyone north of Lake Constance to be able to read it.
The differences between British English and American English are indeed "wildly, insanely overstated," especially when contrasted with the differences between High German and Swiss German. One response to this claim might be that I am contrasting two standard varieties in one case and standard and non-standard varieties in the other. But my hypothesis is that a speaker of Southern American English and a speaker of the English spoken in Scotland would be able to communicate much more while sticking to their two varieties than someone from Hamburg and someone from Basel if they also stuck to their two varieties. Now if only someone would test it.