In 2010, we explored a problem with the "Remember Me" option. In 2011, we had a new twist on the login problems - inability to logout from Blogger.
I cannot sign out of my Blogger account, to access a different Blogger in account. Blogger keeps returning me to the same account.
Since our login state is retained in a cookie, that's stored on our computer (not in the Blogger servers), it's likely that the base cause of this problem, like similar login problems, involves cookie and / or script filters on our computers.
It's also possible that changing your login policy may make a difference - and some account owner confusion may be present.
Epidemiological analysis of the problem, as reported in the forums, suggests a low level of effect - when simply compared to similar reporting levels of the above mentioned "Remember Me" problem. If we consider the possibility that the need to logout, which becomes acute primarily when a blog owner maintains 2 or more Blogger accounts on the same browser, will be much lower than the need to stay logged in, it's likely that this is a more severe problem than what is cursorily visible in the forums.
(Update 5/24/2011 10:30 PDT): Blogger Support has acknowledged a possibly related problem.
We're investigating an issue which is preventing login and comment posting for some users, and hope to have a fix released shortly.