I'm working on a drawing with several shapes. One of the shapes will not make new nodes when I double click on the line. I can create a new node by individually highlighting two nodes - one on either side of the target space - and clicking the "+" on the toolbar, but it alters the path of the line and so isn't a great way to do this.

I'd like to find out why this happens with some shapes, and if there is a way to correct it?

You have a spiro path lpe applied on your path. remove/hide it in the lpe editor or convert object to path (Ctrl+Shift+C). (Ctrl++ can work this case too.)