Yes, any JAR country can issue a JAA license. You might find, though, that a converted license is annotated as such and does not have to be accepted by all JAA countries.
It is also possible to do training in different JAA states, provided the state of license issue has agreed. You might be able, for instance, to do groundschool and exams in Spain, skills test and IR in Italy and get an Italian license, if the Italian CAA agree.