I had the same problem for several weeks. If I wanted to check out ten feeds in a session, I would have to enter the log-in info ten times. It was so frustrating, I purchased Newsclip, so I could continue reading feeds. Then the Bloglines log-in problem went away by itself. Now I'm back to using Bloglines.
Good luck finding a fix -- I feel your pain.
DavidE
www.davidenglish.com