@Mizar
I have what sounds like a similar or maybe the same problem. But note that I am on dial-up still and I thought that this problem was specific to my bad phone line conditions, etc.
Any individual program stops communicating after some time. To keep AWeb working, I leave a page open with small images on it (e.g. a page like this in a forum) and ask to reload a small image when the xfer stops. After a while, I need to start a xfer several times to keep things going.
I haven't tried on my A1200, but this happens with my Amithlon box running Genesis and is also happens on my OS4 box.
A "solution" that works for my OS4 box is to open a command line shell and type "ping
http://www.google.com interval 5" (for example) for the whole time I'm online. Xfers then only stall for five seconds or so, not forever.
And yes, the same problem afflicts my Linux box. It really seems to be something caused by bad line conditions or something at my ISP.