A good way to diagnose connection problems might be to try a URL like this in a browser: browser (click the link) : [http://api.openstreetmap.org/api/0.6/map?bbox=-0.108157396316528,51.5073601795557,-0.107599496841431,51.5076406454029][1]
That What does it tell you? Well if you get an error, then it tells you there's something definitely wrong with your internet access. If you see XML, then it hasn't helped you so much, but it still narrows the problem down a little. This is a worthwhile step to diagnose the problem and worth mentioning that you've tried this when asking people about it.
More technical details: What this actually is, is a direct request to the OpenStreetMap API from your browser, request an area of map data. It should give you back some XML data like in the [example here][2]. This JOSM will do almost exactly the same. Only difference is an API "map call", which is what JOSM is using to download.
the HTTP request will have a different user agent (*possibly* blocked by a firewall) and maybe you're requesting a larger area of the map.
[1]: http://api.openstreetmap.org/api/0.6/map?bbox=-0.108157396316528,51.5073601795557,-0.107599496841431,51.5076406454029
[2]: http://wiki.openstreetmap.org/wiki/.osm#Example
