In Berlin, both semester courses and intensive courses in the breaks are available at the three universities. Both in German and in foreign languages. In principle, these courses are significantly cheaper than equivalent ones offered by other language centers.
The BMS pays one intensive (short-term) German course for you during your whole BMS program: you can either apply for the BMS German course offered in each September, or take another course and ask for reimbursement. It must not cost more than 150€, and you must attend more than 80% of the classes in order to obtain funding from the BMS.
Semester courses are cheap (even available from ~50€/semester), usually 4-6 times 45 minutes per week, in one or two blocks of classes. The higher the level is, the less classes per week you will have. Usually if you take a semester course at your own university, you can obtain some credits for your Master's degree for them. (Check if this is true in the case of your course!)
Break courses are more intensive (usually held every working day of the week), and more expensive. They provide you the same kinds of language level certificates as the semester courses.
One has to be really fast if a language course requires online application!
FU:
FU Language Centre: http://www.sprachenzentrum.fu-berlin.de/en/index.html (cheap, standard, large groups)
HU: Every
HU Language Centre: https://www.sprachenzentrum.hu-berlin.de/en/courses-and-registration (cheap, standard and larger groups during the semester. A bit more expensive intensive courses, but still not too much, compared to other language centres)
TU:
ZEMS: Zentraleinrichtung Moderne Sprachen (cheap, standard, larger groups, one has to be fast to get in. Placement tests are available, but it's not required to do them in order to apply for the courses. Much better offers during the semester than in the breaks) https://www.zems.tu-berlin.de/sprachen/parameter/en/
SKB: Sprach- und Kulturbörse Berlin (less cheap during the semester, smaller groups, more intensive. It's easier to get in but you need to go there for a placement test. Intensive courses not very expensive compared to other language centres): http://www.skb.tu-berlin.de/contao/index.php/en/courses.html
You are also allowed to take courses at the universities where you are not enrolled, but then the course will probably not give you credits for your Master's studies. Students of the mother university always have priority to other students at the applications for the courses.