Plagiarism is the use of others' ideas or distinctive language without sufficient attribution. Your question is about the amount of allowed duplication in your own work. That should (I think) not be talked about in terms of plagiarism. The term "self-plagiarism" is sometimes used for this, but I don't like it: the two academic crimes are inherently different because the victims are different. In a case of plagiarism, the primary plagiarized party is the victim. In a case of improper duplication, the victim is, in a more diffuse way, the rest of the authors' academic community: they are, in several different ways depending on the situation, "paying twice" for the same product.
So "Is this plagiarism?" is (I think) not the right question here. (Answer: not if each paper clearly cites the other and duplicated content is pointed out.) The right question is Is the decision to make two papers out of one project with a substantial amount of identical content serving the academic community well? What aspects of this course of action are potentially problematic, and what else could we do instead?
I would begin with
Our research (me being one of the co-authors) consists of two main sub-cases which are too long to be in a single paper.
How do you know your work is too long for a single paper? At least in my part of academia [mathematics], on the one hand the length of a single paper is quite variable: many papers are published each year which are under 5 pages and over 100 pages, for instance, and relatively few journals have strict length requirements. I have heard that length requirements are more common and more severe in other fields, but I would ask whether you are specifically violating the length requirements of all the best fit journals in your field. On the other hand, the length of an academic paper is quite fungible: in my experience, most papers can be compressed by up to 25% without a reader really noticing a change, and many papers can be compressed by up to 50% without ruining the core content. That amount of compression could solve your problem. Have you tried?
Sometimes people do have feelings that "this paper is too long," but that can be quite subjective. (For me as an author, fatigue starts to set in somewhere between 30 and 40 pages. After 40, the burden of keeping the entire paper in mind as any individual changes are made starts to feel significant. My longest paper is about 50 pages and had a very energetic coauthor.) If you are not actually violating length requirements or very clear norms of your field, I think it would be better to submit a paper which feels a bit on the lengthy side and see what feedback you get. Sometimes referees and editors recommend splitting a paper in two -- if you get such a recommendation, you can be more confident that you made the right decision.
Note also that your choice of "splitting" seems not to be a very natural one: you are not splitting the paper into two parts which can each stand on their own, and because of that you are repeating a lot. This makes me think that even if you have decided to write two papers and not one, a different way of splitting might make more sense. If your paper is unavoidably too lengthy to be published as one, then that should imply that you have enough academic content for two papers. What you have done so far -- repeating several sections verbatim for most papers -- is likely to create the impression among your readers that you do not have enough academic content for both papers but are trying to stretch one paper into two anyway.
Well, I hope this gives you some things to think about. I can't really give you the answer without seeing your paper. Among other things, it depends on the proportions of everything involved. If for instance the "theory, introduction, procedure" occupies two pages, and each case occupies 20 pages,
then the amount of duplication is probably acceptable. Even in this case though a different reorganization might serve you better. In my experience, breaking something unnaturally in half and trying to publish each half separately can devalue your work: academia wants to publish "full things" rather than "half things". However, this is ultimately quite subjective, and with sufficient reorganization you could probably overcome this.