If statements are mutually exclusive, so since you've categorized all of those organization types before "Tim", you'll never find a match for him.
I believe you could write separate calcs for each person to address this:
If [Organization] = "Paper"
or [Organization] = "Cardboard"
If [Organization] = "Plastic"
etc, etc. This is untested and I'm just thinking out loud, but I think this will allow you to use the duplicate dimension members.
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Also, you can enforce row level security by using User Filtering as well. Tableau Server makes this very simple, too, as you can set User Groups, etc.
Thanks so much for your help Matthew. I've been staring at this for weeks, I'm not sure why the simple approach of pulling the formula apart didn't hit me. I haven't attempted User Filtering yet but I think I'll add that to my list of the next things to learn how to do. It sounds like it would be easier to do it that way, I just have to teach myself how to do it!! Thanks again!!
No problem! If you have Tableau Server, it can be VERY simple to set up User filters--you just select which Server users should see what dimension members, add that as a Filter and you're done. That method is shown at the link I provided; there are also User functions in Tableau that allow you to filter based on usernames or groups in your database.