So, many thanks to those people who helped me out with my "noob questions" thread.
I attempted to follow the instructions at tomatousb(dot)org/tut:installing-on-asus-rt-n16, but after an apparently successful upload and restoration, I couldn't connect to 192.168.1.1.
So I left it a few minutes, thinking maybe it needed a bit of time to process something, maybe… and again failed to connect.
Then I followed the process to upload the ASUS firmware and try to get it back to factory so I could start again.
That also seemed to go well, until I failed to be able to connect to 192.168.1.1 again.
So I did a factory settings restore (red button) before putting it into rescue mode again, and this seems to have broken it.
I can turn it on with the black button depressed and after a couple of seconds the PWR light slowly flashes, indicating that it's in rescue mode. It will stay in rescue mode for however long I want it to, as long as I don't attempt to do anything to it, like upload new firmware with the ASUS rescue utility.
Because the moment I start the firmware restoration, the PWR light stops flashing, the utility counts to about 20 and then terminates with status "The wireless router is not in rescue mode."
I am reassured by posts here saying the RT-N16 is fairly difficult to brick, but note that there are plenty here who have managed to brick them anyway. Am I the latest?