Sometimes Google's security features are extra paranoid and block Google's own services. I tried to use the mail fetcher feature from a secondary Gmail account and Google mentioned that the authentication failed (it's been enabled before). I entered the right password and Google still couldn't authenticate. Then Google started to show warnings in my main Gmail account, at the top of Google search pages and even sent an email and an SMS message: "Someone recently tried to use an application to sign in to your Google Account. We prevented the sign-in attempt in case this was a hijacker trying to access your account."
Google sent me to this page which says: "We detected activity on your Google Account from a location you don't usually sign in from." The IP address is 126.96.36.199 (mail-pd0-f147.google.com) and it's from United States. Obviously, it's Google's own IP address.
How to fix this issue? Go to this page, click "Yes" and "Yes - Continue". From the Google confirmation message: "As a security precaution, Google may prevent an application from accessing your account if it's the first time we've seen this application sign in to your account, or if it's attempting to sign in from a new location."
Then Google sends you to this page and you need to click "Continue" and "sign in using the application you want to authorize access to your account within the next ten minutes."
Unfortunately for Google, it wasn't even the first time when Gmail's mail fetcher was enabled. Google should find a way to make Gmail's mail fetcher work without having to jump through hoops.