Speaking anecdotally, I know people with last names like "Miller" who were originally German (last name Mueller).
Miller, Meller and Mueller could as well be Polish people. Miller and Millar are also common surnames in Scotland.
My mother's maiden name is Meller. One of Poland's former Prime Ministers is Leszek Miller.
Here is a good website using which you can check distribution of surnames in several countries:
Mueller in Poland:
http://www.moikrewni.pl/mapa/kompletny/mueller.html
Miller in Poland:
http://www.moikrewni.pl/mapa/kompletny/miller.html
Meller in Poland:
http://www.moikrewni.pl/mapa/kompletny/meller.html
German version of this website is verwandt.de:
http://www.verwandt.de/karten/
German surname doesn't mean you are German, Polish surname doesn't necessarily mean you consider yourself Polish.
For example among the top current leaders of German minority in Poland are: Gerhard Bartodziej, Herbert Czaja and Herbert Hupka.
In case of Mr Bartodziej he has a Polish surname and a German first name. But this name also has a Polish form - Gerard.
My late grandfather was Henryk, if he was Heinrich you would think he was German, as he had a "German" surname.
To summ up - ancestry reports aren't perfect, but they are still a much better source than looking at surnames.
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One of Poland's former Prime Ministers is Leszek Miller. (...) Miller and Millar are also common surnames in Scotland.
I don't know what is the origin of Polish Miller surname and family, but it could as well be Scottish not German:
There was a considerable Scottish settlement in Poland:
http://homepage.ntlworld.com/josephd.ross/index.html
The number of Scots in Poland which Lithgow quotes (30,000 families) is accepted in principle by T. A. Fischer
On the other hand, a lot of Millers can also be found today in Southern Germany (check the map from verwandt.de). So hard to say.
And when it comes to surname Meller - it seems to be much more common in Belgium (Flanders) and in the Netherlands than in Germany.
But all of my Meller relatives' ancestors identified themselves as ethnic Poles as long time ago as the oldest of them can remember.
So I don't know where did this surname come from. I suspect so called Hollanders (Olędrzy), who settled in Poland during the Early Modern Era.