We have been getting this message on our sites and after doing some research I see that it could be that the ip to our host is being blocked. We host with WP-Engine which is an enterprise level WordPress host and have a dedicated IP address so I question if it is that.
Currently we have 30 websites on a Multisite install which only 24 get traffic. We are using the Genesis framework and their default twitter plugin which has the refresh rate set to 5 minutes.
Here is the IP address of our server
66.228.51.93
Here are some of our domains
http://blog.lifeway.com (and multiple sub-directory sites like http://blog.lifeway.com/parentlife http://blog.lifeway.com/christianfiction etc)
http://thomrainer.com/
http://ericgeiger.com/
I am going to test the twitter code vs the plugin, but of course the only way I will be able to see if it makes any difference is when the feed hangs up again. Should we increase the refresh rate? Any other suggestions?