What a fantastic idea. This seems worth pursuing. It should be possible to configure a modern browser to work with low bandwidth: HTML/CSS/JS would load, but images and other media wouldn't. Is there any reason why HN, Reddit, Twitter, webmail, and other services like IRC wouldn't be usable under those conditions?
It seems like people might be much more willing to rent out their infrastructure to anonymous parties strictly for those purposes.
Also, you don't really need to configure a browser - just use lynx, which will ignore most of the bandwidth hungry aspects of a site.
The text on the current top story (the Wright Brothers article) is about 11kbytes. That doesn't include any html or css or anything else. That would make a page load at over ten seconds just for the text.
The point isn't that it can not be done, but that people would not tolerate it unless they had a real need.
I'm not sure if you would count this as a solution, but, conceivably you could "enable anonymity" at very low bandwidth ... say ... the equivalent of 9600 baud ?
This is fast enough for speech. It is not fast enough for any kind of multimedia that would be acceptable in 2014 and beyond. It might be a barrier that would cause all bad guys to use other networks, but still allow the kind of "freedom" that we're all convinced twitter gives us (and so on).