Earlier quoted context omitted.
The application doesn't know if the username you entered is actually yours or someone else's. You could provide the correct password to your account 'test', but not 'ttest'. The server just tells you to check both instead, it's more semantically correct and offers some security improvements with user enumeration.
It knows if it doesn't exist though. This whole thing is about sites that say "Username or password is wrong" when they know the username is wrong because it doesn't exist!
Imagine both 'dave' and 'davr' have an account. I'm 'dave', but I accidentally type 'davr' and my correct password. Now the site will tell me that my password is wrong. So I retype and retype my password over and over again and still can't log in, because the problem isn't the password like the error message says, but rather that I typed the wrong my username.