Pagination Not Working on Category.php page

Try this code, not much different from your’s but with proper nesting. Let me know if that works.

<?php if ( have_posts() ) :
    while ( have_posts() ) : the_post(); ?>

    <article id="post-<?php the_ID(); ?>" <?php post_class(); ?>>

        <a href="https://wordpress.stackexchange.com/questions/212425/<?php the_permalink() ?>" class="noborder"><?php if ( has_post_thumbnail() ) { the_post_thumbnail('thumbnail'); } ?></a>
        <a href="<?php the_permalink(); ?>" title="<?php the_title_attribute(); ?>" rel="bookmark"><?php the_title(); ?></a>

            <section class="entry-content">
                <?php   $content = get_the_content();
                        $content = strip_tags($content);
                        echo substr($content, 0, 250) . '...';
                ?>
                <a href="https://wordpress.stackexchange.com/questions/212425/<?php the_permalink() ?>" rel="bookmark" title="Permanent Link to <?php the_title_attribute(); ?>" class="button">Full Post</a>
            </section>

    </article>

    <?php endwhile; ?>
        <nav>
            <div class="6u"><?php next_posts_link(sprintf( __( '%s', 'blankslate' ), '<span class="button">&larr;  Older Posts</span>' ) ) ?>
            <div class="6u"><?php previous_posts_link(sprintf( __( '%s', 'blankslate' ), '<span class="button">&rarr; </span>' ) ) ?></div>
        </nav>          

<?php else : ?>

    <p><?php _e( 'Sorry, no posts matched your criteria.' ); ?></p>

<?php endif; ?>