Are you positive that you have to sign into the PlayStation Network in order to access Netflix? Netflix streaming should be based off of an Internet connection, which itself should be acceptable without having your PlayStation 3 account logged in. If you want you could always terminate the automatic sign in so that step is skipped. Then all you would need to do from there is access Netflix, assuming that there's no problems with the Internet connection.
You don't need to sign in to PSN to use Netflix. It worked even during the PSN downtime (it asked a few times though). I'm always signed in so Netflix is not in any way inconvenient; the app loads and there are the videos. I've only had to update (PS3 or netflix) twice since they started offering it. The CD was much less convenient for me.
You dont have to be signed-in to the PSN to use netflix....
If you're having connectivity issues try resetting your modem and router...... I've never had an issue when the PSN was under maintenance. They dont take down all the servers at once ya know, so people can STILL log-in no matter what part of the country you're in. It just might take a little longer