Glad someone stumbled across the Cookie thing!!!
I'm not sure if this works on all browsers, but on Firefox I just searched through the cookies for gog.com, and deleted only those, also cleared my cache (as someone had suggested it) then exited the browser and relaunched. Seems to have fixed it!
I'm having a I guess separate problem with downloads failing 90% of the time on Comcast, but I guess it's unrelated :-/
Cyker: Confirmed: Clearing cookies fixed the login problem for me.
It seems GoG at some point changed its cookies and I also had to allow them for for gog.com, www.gog.com and login.gog.com, whereas gog.com was previously enough.
Clearing the cookies caused it to ask instead of silently fail and this seems to have done the trick.
I think I have also sorted out my cart problem, but it required me to change the webstorage limit for gog from 5MB (!) to 20MB (!!).
The limit was only 1MB a few years ago!
What browser are you using that it asked you that, out of curiosity? Firefox has an "offline web content and user data" section, but the only sites using it so far are 2 Amazon sites and Bing, using a total 6.7MB.