About a week ago I moved two platies to the "hospital" tank. A male that was looking really bad - he had significant tail rot, and a female that wasn't looking too good. I treated the tank with Melafix, and hoped for the best. The female died a few days later, but the male seems ok. His tail hasn't healed yet, but the large whitish spot at the base of his tail (which he has had for several weeks) is almost gone.
I believe that his main problem was bullying - he was the second largest male, and had been harassed a lot by the largest male. While that fish has almost doubled in size since I have him, this male hasn't grown much. Until today the male in the hospital tank hasn't looked very happy - he spent a lot of his time sitting near the surface with his fins clamped - today he's much more active, poking around the bottom of the tank (and finding something edible, by the look of it). Hopefully he's finally on the mend. I still think his main problem was social, not water quality or disease per se. Hopefully I'll be able to get a second tank this weekend and move him into it. It might be worth making it a male-only tank (for platies, at least) - put him and the two smaller males in there - without females to fight over they may get on better. Probably have to move the male babies across there too.
Of course, this still leaves the question of what I want to do with the platies.