Update (again): While resetting the settings did solve my problem temporarily, I had the same problem the next day as well. The only meaningful thing that I had changed was the global maximum number of connections (Preferences -> Bandwith) from 150, which seemed excessive, to 15. I changed the value back to 150 and it started downloading immediately! This is strange, since the value was 15 for years, I think. Well, I am getting tired of these solutions that only work until the next day. Hopefully this is the last time the problem appears.
Update: The enthusiasm I had after making some of the torrents work faded when I noticed most of the other torrents were still not working. My only solution was to go to the Application Data folder, then Bittorrent, and delete settings.dat and settings.dat.old while the program was closed, then restart Bittorrent. Warning! You will lose all the torrents and settings so save them first (I selected them all and copied the magnet links, then added them back one by one afterwards). The weird thing is that, while it worked, the software also looked quite different from what I was used to. Perhaps just blindly updating the version of Bittorrent all the time is not the best option. Sometimes we need to reset the setting to take advantage of the new software settings.
I've had a problem with a router which prompted me to remove it and put the Internet cable directly into my computer port. I wouldn't recommend it for you, since a router adds a level of protection against outside access, but still, I was not home and a friend of mine "fixed" the problem. Since then, I couldn't download anything on Bittorrent, as it got stuck at "Connecting to peers", even if it connected to the trackers, and saw all the seeds and leechers. Googling around I found a suggestion to remove "resume.dat" from the Bittorrent Application Data folder, but it didn't work for me. I tried a lot of things, but none worked until I found an obscure forum asking about a proxy server. And it dawned on me that it could be from there.
You see, I had previously installed a free proxy program called Privoxy, but I had bound it to the previous IP address of the computer. As a result, it didn't load anymore. The Internet settings were OK, I had removed any reference to a proxy, but strangely enough something in the Bittorrent client kept some information relating to the proxy. After I only bound the proxy to the local address, Privoxy started and miraculously also did the Bittorrent downloads (after a restart of the program).
So, if you have issues with the Bittorent client getting stuck at "Connecting to peers", check if you have recently changed your Internet proxy settings.