Thanks to David Pashley, I've now learned the problem. Vodafone transform some of the content they deliver over the web. One effect is degraded image quality (discussion of the problem and its solution in Germany is here and here). The other problem is their use of minify, which is breaking some sites. This was breaking the JQuery library included in a site of David's I use.
We resorted to loading the libraries over HTTPS, which is not a good general solution but works in this case. Meanwhile I thought I'd make this entry in case it provides a pointer for anyone facing similar problems.
(Aplogies for the technical post. If you've read this far and are disappointed, here a link to a kitten video —> kittens )