OK, I think I may have solved the problem. After a little more Googling and some further frigging about, I realised that it may be an issue with the driver for my ethernet adaptor.
I have a Realtek RTL8111/8168 ethernet adaptor in my motherboard, and it turns out that there are some problems with the out-of-the-box Ubuntu driver for that adaptor. Happily, there seems to be a different adaptor that can be installed via the repositories that's been designed specifically for that adaptor and that should, in theory, solve some of the problems. Installing it was dead easy, with a single line at the terminal:
It seems to be working for now. I'm not marking this thread as solved just yet: I just want to make sure that it remains working consistently over the next few days.
sudo apt-get install r8168-dkms