I've 3 classes in WordPress (the question is unrelated into it):

class WP_Widget

class Theme_Widget extends WP_Widget

class Specific_Widget extends Theme_Widget

Basically Theme_Widget consists of some extension functions towards the fundamental Wordpress_Widget.

Inside Specific_Widget I call among Theme_Widget's techniques:

class Specific_Widget {

    function __construct() {
         $this->some_method_that_belongs_to_Theme_Widget();
    }
}

After I instantiate Specific_Widget, PHP throws a fatal error the following:

Fatal error: Call to private method Theme_Widget::some_method_that_belongs_to_Theme_Widget() from context 'Specific_Widget' in ...

Have you got a concept regarding the way i can resolve this? This is actually the very first time I have received this error from PHP. Can it be be a consequence of WordPress itself?

You have to declare your method protected, instead of private, if you want child classes to have the ability to utilize it.