Perhaps it is too early to ask this, but I will go ahead anyway, to satisfy my curiosity. I am currently studying Arts in a University, aiming to get a BA Honours in English and German. If I went ahead to get a Masters, would I have to choose one of them, or could I continue with both? And similarly for a Doctoral, if I get that far, does it have to be one subject at a time?
If you do research based Masters and/or PhD, you could probably find a specific research that combines elements of both - quite often (not always), these types of projects turn out to be the better ones, as they force the researcher to establish, rationalise and justify the links between them.
Even though my research has not been in linguistics, rather in the applied physical sciences field, my research combined a few subjects together - in my case: atmospheric physics, programming and signal processing.
So, it is very possible to do, in your case, relevant elements of both English and German in a single research postgraduate degree.
If you're question is:
Can I do a 'double master's' in English and German the same way I can 'double major' in English and German?
The answer is not easily. The reason for this is that admissions to master's programs are normally decided by the department, not by the university as a whole. Therefore, admission to one program (say, a master's in English) does not give you the right to get a master's degree in the other program (here, German). You would have to apply to the programs separately. It is also difficult to study for multiple master's degrees simultaneously, since the coursework tends to be disjoint sets, and therefore you would probably need to complete the degrees sequentially (at best).
The doctoral thesis topic would allow you to combine the two disciplines. However, it may be administratively difficult to have two different departments listed on your degree; this is an administrative issue rather than an academic one. There is usually an approved list of fields that can be listed on the diploma; if you want something else, you will have to petition for it, which could be a difficult process.