There are several German words whose English cognates have a more specific meaning than the original word.
German 'sterben' to die ---> English 'starve' to die from hunger
German 'Tier' animal ---> English 'deer' a specific animal
German 'Hund' dog ---> English 'hound' a specific type of dog
German 'Bein' leg ---> English 'bone' part of your skeleton (i.e., the thing that's in your leg) -- this one seems to be an interesting case because according to my dictionary it means both 'leg' and 'bone' in German
That's all I've got. Does anyone else know any other cognates like that or cognates that are interesting for any other reason?