How to display blog posts only authored by the administrator

Four changes to your script. Limit $users to users with ‘administrator’ role. Removed line if($user->caps['Administrator']!=1) continue; as every user is an administrator. Changed ‘post_per_page’ to -1, to process all posts. Added wp_reset_postdata(); after each loop:

$users = get_users( array( 'role' => 'administrator' ) );

foreach ( $users as $user ) {   
    $query = new WP_Query( array(
                            'posts_per_page'=>-1, 
                            'author' => $user->ID
                            ) 
                    );
    if( $query->have_posts() ) {
        while( $query->have_posts() ) {
            $query->the_post();
            get_template_part('content', 'postlist'); 
        }
    } else {
        get_template_part('content', 'none'); 
    }
    wp_reset_postdata();
}

Make sure that you reach right template in get_template_part('content', 'postlist'); statement.