If you're developing on localhost, may be your firewall blocks the apache http.exe server.
If your site is hosted on a web server, and you see the error message 'An Unexpected HTTP Error occurred during the API request' after requesting a page on WordPress Admin Dashboard and you are getting this error frequently on your dashboard, then it means you have to increase the wordPress timeout limit for every request. To do that
- Open your FTP Client software
- locate the file "class-http.php" inside "wp-includes" folder.
- And find the code "'timeout' => apply_filters( 'http_request_timeout', 5)" and change the timeout limit to 30. (5 --> 10)
- Make sure your connection is active (probably you checked already)
- And make sure server software in not blocked. In my case, it is httpd.exe (Apache HTTP Server) This is only required if you're accessing to the internet for something like plugin installation and, etc....