I saw these messages a few days ago on several of the WordPress sites I own or manage:

RSS Error: WP HTTP Error: Couldn’t resolve host ‘wordpress.org’
RSS Error: WP HTTP Error: Couldn’t resolve host ‘planet.wordpress.org’
RSS Error: WP HTTP Error: Couldn’t resolve host ‘blogsearch.google.com’

It took a great deal of Google searches to finally narrow this down to a hosting issue. Now as I often recommend HostGator as a hosting service for WordPress installations (or any site for that matter) I contacted @HGSupport on Twitter to see if this was possibly a widespread issue:

Which eventually lead to creating a support ticket (#EBL-19977486) so HostGator could review all of the sites I own. They finally responded with:

Basically there were 2 issues effecting the feeds from displaying. First, we needed to make changes to the RLimit for the user, then clear the transient cache out of the WordPress database as it will continue to display a cached error after the RLimit is updated.

This resolved the issues I was seeing with my sites but HostGator is treating these reports on a case-by-case basis. That being the case, if you are seeing similar issues with your WordPress installations and use HostGator, please contact them as soon as possible. Feel free to note the support ticket created for my sites if you feel it will help with your situation. If you are using another hosting service feel free to note HostGator’s method of resolving this issue.