I am using Google Webfonts on my website. In the console on Chrome I get the following warning:
Blink is considering rejecting non spec-compliant cross-or
Ok, so clearly Google Fonts allows cross-domain access or their service wouldn't be much use.
The explanation for my particular problem is that I'm on a corporate internet connection. It must use some proxy that is stripping out access control headers from web responses. This causes the browser to give warnings or failures when loading the resources.
I've contacted IT warning them about their misconfigured proxy and hopefully this will be fixed.