I couldn't get into twitter, messages or facetime.  What fixed it for me was checking the automatically date and time

At the top right click the day and time --> Choose "Open Date & Time Preferences," and make sure that "Set date and time automatically" is checked.

Try this, it worked for me:

System Preferences --> Mail, Contacts, & Calendars --> Remove your Twitter account by clicking on the at the bottom of the left hand pane in the window.

Go to your Twitter account and under your Settings --> Apps, revoke access to OS X Integration and click 'Save'.

Return to your System Preferences where you removed your account and attempt to add the account again. After you enter your @username and password, attempt to use the app in the notification window.

It fixed the problem for me and even gave access to the OS X Integration back to my apps on my Twitter account.

Prior to this, the Mail, Contacts, & Calendars would tell me that it was 'Unable to verify account username and password'. Very frustrating.

I hope this helps you!

I did this for my gmail account - not luck. I removed both my gmail accounts and tried to add them back - it still says "Unable to verify account name and password".

Did you try to add your device in gmail security settings? I had the same problem and it helped.

1) Go to Ur Gmail a/c
2) right side top , Privacy (it will take u to ur gmail Accounts)
3) go to Security

4) Connected applications and sites > manage access > enter ur password
5) then Generate APPLICATION SPECIFIC PASSWORD for Apple mail
6) Now Enter that password while Entering in ur MAC (dont enter ur regular GMAIL Password)

Ta Daaaaa !!!!!!

Same problem with Yahoo! account. I couldn't set it up in Settings, and then set it up from Safari, but Mail and Settings keep on asking for the password.

Try to create new user account on your Mac — it helped me with logging in to twitter/gmail and facebook.

Hey guys,

I think iPraj is right -- or at least that's what worked for me. You have to create an application-specific password in your Google account and use that in the account setup in the System Prefs (not your actual Gmail password).

It's the same process I had to use to log in on my Apple TV; it's because these products can't accept the text message that is part of the two-step verification process.

Follow the instructions above on your Google account and enter the password that's generated in your System Prefs when setting up the Gmail account.

This happens to me as well. It says my password is wrong:

unable to verify account name and password

In red letters, and I am 100% sure the password and account (email) are correct! Things like this make me abandoning Apple altogether, and I am a big fan. They don't seem to work as good any more.

You might want to consider starting a new discussion. Since this one is a couple of years old, less people are likely to look at it. You can link to this one.