I have combed through the site and the only solution that I can find for this is to extend the timeout to 30 -- and I have done that and it didn't work. I have also uninstalled and reinstalled every plugin, switched to the default theme, etc.
I have contacted my host and they seem to think the problem is with WordPress. I am really scratching my head here.
I have two sites and they are BOTH spitting this error when I try to search for or update new plugins:
An Unexpected HTTP Error occurred during the API request
Additionally, on the dashboard, the WP development blog feed and the new plugins feed both also spit back errors or never load. They pop this error:
WP HTTP Error: couldn't connect to host
Both sites were fine up until a couple of days ago and then everything just blew up. I didn't change anything, so I don't know what caused it.