Does Java really support multiple inheritance or not?

Yes and no. The fact is that Java separates implementation inheritance from interface inheritance. Interface inheritance deals with the "contract" which a class adheres to while implementation inheritance deals with the reuse of code and the contract which a class adheres to.

Java only allows "multiple inheritance" of interfaces. It does only allows single inheritance of implementation. This is a good bit different from e.g., C++ where both concepts are intermingled (which I think is quite ugly and needleslly complex).