In the past twenty-four hours, it seems as if Google sent out a message to thousands, maybe hundreds of thousands, of site owners, that there is a problem with their site.
Typically, you never ever want to ignore a webmaster tools notification but in this case, the messages were sent in error.
Google told these webmasters that "Googlebot can't access your site", that the site had connection failures and Google was unable to access your site.
Here is a sample message one of my clients received (although I received several):
Google's Matt Cutts wrote in the thread that this seems like an error on Google's side. He said:
Hey everyone, please don't worry about this message at this point. Enough people are getting this message that I suspect it's an issue on our end.
I've got an email out to our webmaster tools team, and we'll figure out what's going on. Thanks.
John Mueller of Google is asking webmasters to post example URLs so they can fix the issue.
The thing is, I checked this morning and webmaster tool's console does not show the error on these sites still, so maybe it was a Google error.
Forum discussion at Google Webmaster Help.
Update: This was indeed a bug, John Mueller later on said:
From my information, this should be working again in the meantime. We will continue to monitor the status and I'll update here should anything change.