TL/DR: exiting CP and re-opening it fixed the issue. But no idea why it happened in the first place.
Using CP4 in part mode. Earlier today, for some reason, I couldn't drag features in the model tree to re-order them or the Insert Here to activate Insert Mode. I could select them, and RMB worked. And Reorder and Insert Mode worked through the Operations menu. Pretty darn inconvenient. And frustrating for a while. Ultimately I exited CP and re-launched. Then it was working again.
Any idea what could have happened to break dragging in the model tree?