My father is Norwegian (Lutheran) while my mother was German, half jewish and half catholic. So my maternal grandfather was a German Jew, this qualifies his DNA as Ashkenazi. Apparently this group was very inbred to the point where I am finding many many matches at 23andme for my DNA (from him) that are too far back to find the paper trail. And we have most of our jewish ancestors documented back to the late 1700s. To be specific, of my over 1000 matches at 23andme at least 900 or more are Ashkenazi of which over half are Eastern European. As we have no known Eastern European family [correction: one 4th grandfather born in Ukraine plus another unknown], I have to conclude that siblings of our ancestors in the 1500s and 1600s migrated East [correction: much movement back west in the 1600s for example Sulzbach welcomed back eastern jews in 1666]. Our Floss families came there from Austria which deported its Jews in 1420 and again in the mid 1500s according to the Wikipedia article on Austrian Jewish history. Probably at that time other family members went further east.
So far I have found one new 5th cousin through the 23andme site on the jewish side (and that was from a surname scan) as compared to about six new Norwegian relatives. And it seems that the Ashkenazi matches drown out our other German matches. Of course the problem may also be that very few Germans have tested since it is apparently not legal there. Fortunately lots of Norwegians have tested.
Found a couple of interesting URLs on this topic: