At some point I realized, that the browser of my HTC Magic does not open pages, accessible over HTTPS (like the login screens for most web sites, for example). I was surprised, because I remembered that I have been able to open the same pages before, on the same device. I tried to delete cookies and clear the cache of the browser with no luck – those pages did not load. The browser seemed to be trying to load them, but it was only displaying a blank screen – no error messages or other pointers that there is a problem.
I wondered what was going on for awhile, and then it occurred to me. I was using internet over WiFi intensively before, and lately (since I have a “fat” data plan) I am mainly using the mobile network. I tried the same pages over WiFi and they loaded perfectly fine. It was obvious that the problem had something to do with the mobile network connection.
I googled the problem and almost immediately came across this link: SSL web sites fail to load in browser application when not connected via Wi-Fi. The problem resided in the configuration for the mobile network data connection. The solution turned out to be pretty simple – from the Access Point configuration (on Android go to Settings -> Wireless Controls -> Mobile Networks -> Access Point Names) clear the fields Proxy and Port, and the problem is resolved.
So keep this in mind, if you experience similar problem – I guess it probably could occur on non-Android devices, as well.