This is the second this happens to me in a week. I buy a new game, try to download it through gog galaxy and get an error message "failed to connect to GOG.com servers". Curiously enough, when I try to download games I purchased a long time ago, it works fine. Weird problem. Did anyone else encounter this?

Yeah, it's a common issue. Not sure what is causing this.
I've had this myself many times, but for me, it seems to be working much better since a few months back.
So, seems like they are improving this slowly.

If a game can't be downloaded through Galaxy straight away with after purchase, you some times just have to wait a few minutes, but it can also be that you have to wait a few hours.
Though with that said, it allways works through the website.