It is only allowed in cases where the person did not deliberately set out to acquire the foreign citizenship. The main examples of such cases have been women who married men from countries like Iran where a wife automatically acquires her husband's nationality whether she asked for it or not. However, such cases are rare and in other cases Japan does not usually allow dual citizenship. If a Japanese national has acquired citizenship of another country he or she is expected to choose which one to keep - Japanese or foreign - and likewise if a foreign national wants Japanese citizenship - something which is quite hard to get, and based on individual cases - he or she must also be ready to give up his own nationality.
If you are born in Japan to parents of different nationalities, you may have citizenship of both countries but will normally have to choose one of the other when you grow up.

You can learn more about these rules here.