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Not hardly. All languages drift over time. Even in this modern age of worldwide mass media, this happens.
Linguists figure that West Germanic broke off from the Germanic language root sometime around 1AD. All West Germanic languages (including the ancestors of English, Dutch, and German) were mutually-intelligible dialects until sometime between the third century and 700AD.
The branch from this root that eventually became modern German was Old High German, which is thought to have become its own language sometime around 500AD. After about 500 further years of slow change it became different enough that we consider it a new language, called Middle High German, and then after 300 more years Early New High German, then after 300 more years New High German (what we know today as "German").
Don't let the namings fool you. Modern German has no more special status as a descendant of Roman-era Germanic than any other Germanic language. In fact, languages tend to change the least in smaller relatively isolated communities, which means you'd expect the closest living language to Roman-era Germanic to be something like Icelandic, which some in fact claim is the case.