Thanks to some people on the DNN Forums, I discovered that the Windows login error “Login Failed. Please remember that passwords are case sensitive” can also mean “you typed your Windows password correctly, but it’s shorter than the DNN default minimum password length of 7”.
The situation was confusing, because some people could log in with their Windows credentials, but others couldn’t.
Either ask your user to change their password to something at least 7 characters long (which is obviously the right way to go), or change the default minimum password length in web.config. The parameter there is “minRequiredPasswordLength”.