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If you repeatedly get bounced back to the sign-in screen, with no message about an incorrect username or password, this is caused by your browser not retaining your sign-in information correctly. You may also see 'Session Expired' messages in these cases. To fix this you need to troubleshoot your browser, so try the following:
Clear your cache.
Delete your Blogger cookie.
Check your cookie settings.
Quit your browser and restart it.
Use another browser like Chrome or Firefox.
Browse Happy—all the cool kids are doing it.
Note that if you have a firewall, you should check its cookie settings as well and make sure it will allow Blogger's cookies to be set. Check the manual or documentation for the firewall for information on this.