Drupal: HTTP Server Request Fails - Solution

The error message is:
Your system or network configuration does not allow Drupal to access web pages, resulting in reduced functionality. This could be due to your webserver configuration or PHP settings, and should be resolved in order to download information about available updates, fetch aggregator feeds, sign in via OpenID, or use other network-dependent services.

There is a lot of answer on the Internet how to resolve it more and less understandable...
Looks like the drupal_http_request_fails flag is set TRUE (1), and Drupal doesn't change it into FALSE (0).

My quick suggestion: direct reset the flag in the database
It is not the real solution, it is not a patch, but works.

Log in into Drupal database
Find the table: variable
Find the row: name = drupal_http_request_fails
Change the value: OLD value =  b:1; NEW value = b:0;
Save it and log out from database.