One of the difficulties in answering your question is that there isn't really any clear consensus on how to categorize case particles in Japanese. It's problematic because the standard conceptual grouping of grammatical case can lead to vague definitions which aren't quite satisfactory. For example, Tsujimura (1996) lists only five pure case particles: が (Nominative), を (Accusative), に (Dative), の (Genetive), and includes the topic marker は as a case marker. Obviously the inclusion of は doesn't conform to case markers in Indo-European languages.
You could argue that other Japanese particles like で are case particles ('Instrumental Case), but Tsujimura disputes that, claiming that case particles don't contain "specific semantic content" whereas で does have that content. The difference is highlighted by the argument that you couldn't acceptably omit で in casual speech, while case particles can be omitted. Tsujimura also disputes まで ('Terminative Case') on the grounds that it can immediately be followed by a nominative case particle, meaning that is should be categorized as a postpositional particle rather than a case particle. Are other particles like より truly Ablative case? If so, how does it differ from から? The definitions are unclear.
As you see, there is no simple categorization which everyone agrees on because it all depends on how to define it, and there seem to be differences in how scholars define it as applied to Japanese. I think we have to just wait for the scholars to argue it out among themselves before we can say definitively what the number is.
Source: An Introduction to Japanese Linguistics (Tsujimura, 1996)