Im sorry Vector, but no, you shouldn't theoretically be able to understand German. They're not that close.
Really? See, I find I can easily understand most simple Spanish and Italian (though to be fair I also speak French and studied a very little Spanish), and people keep going on about how similar German is to English and how easy it is to understand even if you've had absolutely no contact with the language.
I officially disagree :I
If I had to guess, I'd say those people are either schooled in German (presuming they are native English speakers, otherwise it's just the other way around) and therefore have enough knowledge to see the similarities (like I can between Swedish and English), and as it's hard to unlearn or unsee what you have learnt/can see they assume it's equally clear to everyone else.
Otherwise, they don't really know what they're talking about, and probably just furthering whatever clichés or platitudes they've been told previously. German and English are not similar enough to "instinctively" understand without training.
You have a better chance understanding Swedish.
Oh, huh. I shall keep that in mind.
I may not be the most unbiased person in this question (as I said above, it's hard to imagine how it would be without knowing both languages, the similarities and in which way's they differ from each other), but I really feel Swedish and English are very alike. Many words are almost exactly the same, with just a few vowels and/or (mostly or) consonants differing. And that's even without counting the many Latin derivatives that exist in both languages.
But then again, it's probably just because I know enough to the most or some of the habits of the languages as well.