I am having the same Issue. I tried using Chrome and IE. I tried from my work pc and home pc.

My current browser is Chrome 20.0.1132.57 m

I went through all the steps as above. I am using Firefox 13.0.1. my user agent is "Mozilla/5.0 (Windows NT 5.1; rv:13.0) Gecko/20100101 Firefox/13.0.1". I am connecting to BT broadband service via a BT wireless home hub, 36 Mbps.

I have successfully connected to LinkedIn though. I also tried via Chrome, no more luck with FB.

It may be my facebook account: I find it very opaque and can't be sure my set up there is right - FB doesn't work like my brain, hence wanting to post from WP which does... I will probably look for a work-around rather than struggle on, many thanks for the support though! carys

Have you tried disabling all browser extensions and add-ons? If not please try that.
Is your browser set up to be "in cognito" ie. in private browsing mode? If so, disable that setting and try again
Are you using the secure https:// log-in? If not, please try it.
http://en.support.wordpress.com/https/

These are the troubleshooting steps we take whenever any issues like this arise:
clearing your browser cache and cookies > http://www.google.com/support/accounts/bin/answer.py?hl=en&answer=32050
using the browser with all extensions or add-ons disabled
updating the browser to the latest stable version (not a buggy BETA browser version) http://browsehappy.com/
confirming third party cookies is enabled http://support.google.com/accounts/bin/answer.py?hl=en&answer=61416
confirming JavaScript is enabled http://www.google.com/adsense/support/bin/answer.py?answer=12654
confirming Flash is updated http://www.adobe.com/software/flash/about/ and if required installing the latest version of Flash http://www.adobe.com/products/flashplayer/
resetting your router and modem and rebooting your computer
then logging in using the secure https:// log-in see http://en.support.wordpress.com/https/
trying another browser

If those do not work for you then we need to know
(1) how you are connecting to the internet and to WordPress.com. The way you connect to the internet (mobile, satellite, DSL, dial-up) and to your blog, and how many proxy server jumps etc. it takes to connect can cause problems. There maybe be problems with your ISP and/or with the proxy servers. Also note that the main dashboard isn’t designed to work with touch devices through the tablet and smartphone browsers.

(2) exactly which browser and version of it are you using. If you don’t know click here to find out > http://whatsmyuseragent.com/