How can I limit the amount of characters used for description in the manage categories grid?

That field is not filtered in template.

One way would be to filter get_terms() on that page (but I am not entirely sure that won’t break something):

add_action( 'admin_head-edit-tags.php', 'admin_edit_tags' );

function admin_edit_tags() {

    add_filter( 'get_terms', 'admin_trim_category_description', 10, 2 );
}

function admin_trim_category_description( $terms, $taxonomies ) {


    if( 'category' != $taxonomies[0] )
        return $terms;

    foreach( $terms as $key=>$term )
        $terms[$key]->description = substr( $term->description, 0, 50 );

    return $terms;
}