How to filter posts that belong to a specific category only if that is the only category

You will need more than one query because before querying posts you have to know what posts to exclude.

I personally don’t see anything bad in ‘cutoff’. This code should work properly:

Update: It breaks pagination (see comments).

<?php
$category_to_filter = 11;
while ( have_posts() ) : the_post();
    $categories = get_the_category();
    if( in_array($category_to_filter, $categories) && count($categories) > 1 ) {
        the_title();
        the_content();
    }
endwhile;

Update: Following code will not break pagination:

<?php
$category_to_filter = 11;
$posts_in = array();
while ( have_posts() ) : the_post();
    $categories = get_the_category();
    if( in_array($category_to_filter, $categories) && count($categories) > 1 ) {
        $posts_in[] = $post->ID;
    }
endwhile;

$my_query = new WP_Query( array(
    'post_type' => 'post',
    'post_status' => 'publish',
    'post__in' => $posts_in
    )
);

while ( $my_query->have_posts() ) : $my_query->the_post();
    // your template stuff here
endwhile;
wp_reset_query();