Wordpress - Exclude post ID from wp_query

The parameter you are looking for is post__not_in (kaiser has a typo in his answer). So the code could be like:

<?php
$my_query = new WP_Query(array(
    'post__not_in' => array(278),
    'post_type' => 'case-study',
    'paged' => $paged,
));
while ($my_query->have_posts()) : $my_query->the_post(); endwhile;

WP_Query post__not_in documentation

I suppose this was heavy, but to answer your original question, I've collected all of the posts id's in an array in the first loop, and excluded those posts from the second loop using 'post__not_in' which expects an array of post id's

<?php
$args1 = array('category_name' => 'test-cat-1', 'order' => 'ASC');
$q1 = new WP_query($args);
if($q1->have_posts()) :
$firstPosts = array();
    while($q1->have_posts()) : $q1->the_post();
        $firstPosts[] = $post->ID; // add post id to array
        echo '<div class="item">';
        echo "<h2>" . get_the_title() . "</h2>";
        echo "</div>";
    endwhile;
endif;
/****************************************************************************/
// array of post id's collected in first loop, can now be used as value for the 'post__not_in' parameter in second loops query $args
$args2 = array('post__not_in' => $firstPosts, 'order' => 'ASC' );
$q2 = new WP_query($args2);
if($q2->have_posts()) :
    while($q2->have_posts()) : $q2->the_post();
        echo '<div class="item">';
        echo "<h2>" . get_the_title() . "</h2>";
        echo "</div>";
    endwhile;
endif;
?>

The first loop displays all posts in a category, and collects the post id's into an array.

The second loop displays all posts, excluding posts from the first loop.

You have to define the post__not_in arg as array. Even for a single value. And please don't overwrite global core variables with temporary stuff.

<?php
$query = new WP_Query( array(
    'post_type'    => 'case-study',
    'paged'        => $paged,
    'post__not_in' => array( 1, ),
) );
if ( $query->have_posts() ) {
while ( $query->have_posts() ) {
    $query->the_post();

    // do stuff

} // endwhile;
} // endif;
?>