What happens when you try to add another Gmail account to your phone? Are you referring to having two Gmail email accounts on your phone to receive emails?
Do you have 2-step verification enabled on the other two accounts that are giving you an issue?
Try signing into one of the other email accounts on your computer right now. Are you able to sign into it on your computer?
Are you using the stock email app?
Let's try an isolate one of the accounts that isn't working.
On that account (you can pick which one), click here: https://accounts.google.com/b/0/displayunlockcaptcha
Click on Continue to allow access to your Google account
Now try adding that account to your phone. If that does not work, on the same email address:
Go here: https://www.google.com/settings/u/0/security/lesssecureapps
Select Turn On
Now try adding the email account.
Go to system>apps>email and force Stop and Clear Data.
Restart your phone and try to add again
Try downloading the app called K9 mail
Open K9 and see if the Gmail accounts can be added there
Yes, it is a 100% safe app. I would never advise my customers to download something that was not safe. K9 mail is a very popular app.
Please download and try adding it there. I would like to see if it adds within that app.
Yes, you can reply back here at any time and I will be alerted. Please reply back tomorrow when you are free again.
It is nice to hear back from you. Do you still have the same issue?
Thank you for confirming. Since the issue continues through K9 with the exact same error message then it has to be either the username or password ***** is being entered incorrectly.
Are you typing the full email address or just the part before @gmail.com? Try both ways.