Actually the behaviour should still be the same.
The only reason for it to break down that comes to my mind is when the name has been edited in the Layers Panel.
Thanks guys. I deleted the text I had been duplicating and started over and now updating does take effect. I can't figure out what I did unless I accidentally edited the name in the layer panel. Sorry I brought it up, but least I know my system's ok.
Christoph probably has it right. If something caused the name to change at some point - even possibly duplicating the layer - it might stop updating. Though I don't know specifically how it works, I can imagine Photoshop comparing the text string in the image to the text string in the layer name, and only updating the latter if they exactly match.