I suspect this means that the MusicBrainz server is failing to respond correctly. It is perhaps giving an HTTP error 503 to Picard. One way you can check this is to use Picard’s menu command “Help… View Error/Debug log”. (Taken from Picard on Mac OS X.) This shows you a dialog box.
On my computer, when I see those red CD icons, I also see an error message like,
E: 16:04:42 Error downloading http://musicbrainz.org/ws/2/release/5335ef7b-8193-448b-8b7e-62d692923238?inc=release-groups+media+recordings+artist-credits+artists+aliases+labels+isrcs+collections+artist-rels+release-rels+url-rels+recording-rels+work-rels+recording-level-rels+work-level-rels+tags - server replied: Service Temporarily Unavailable
My workaround is to select the CD icon, and click the “Lookup” button. This makes Picard try again. [Instructions from memory; I can’t reproduce the server problem in Picard right now.]
The servers will remain unstable until they are moved to a new host. It’s being worked on, but if you want to do your tagging now, you could right-click on any release with a red disc icon and click “refresh” (or is it reload?) to try again. It usually goes through after one or a few tries.
I note that all the errors from your screenshot contain Cover Art Archive URLs. This is consistent with the older screenshot, where you did get some information about the releases – apparently just fetching the covers did not work.
The CAA servers did have a spell of dizzyness some days ago. I think it passed, but as @mfmeulenbelt mentioned, most ’brainz servers are not totally reliable at the moment.
Personally, I’d not lay off tagging just because there are some releases where you’d have to retry. Remember that, once your files have MBIDs attached to them, loading them into Picard at a later date (for example to get covers or information that was missed in the first run) is quite hassle-free.