I had problem to sign into Google account on my Android OS @ first start of my recovered mobile (SE Xperia X10 mini pro). I had enabled double authentication for my account. Probably this is why I couldn't log in...
So solution which worked for me is: I went to https: // google. sk/ accounts - there are spaces, coz I am new user and forum would not allow me to post links due to spam. (this said login screen when it said bad username or password... it said for recover your account go there... I went. But I was sure my password was absolutely correct)
I added additional email address (address of another registration example: outlook.com). After this I was able to log on with my additional email address to my google account.
So I recover my amdroid to factory settings again and then I used my alternative log in. And it worked! I was so happy, coz I went through 2 hours troubleshooting about my wifi, 3g, another google accounts... and so on
So I hope it will help some1 with same problem.