Breaking Bootstrap Grid at Post Count with Tax Terms

I’m not exactly sure where you want your clearfix to come in, but you should be able to do something like this:

<div class="row">

    <?php

    $libargs=array(  
      'orderby'           => 'name', 
         'order'             => 'ASC',
        'hide_empty'        => 0,  
        'exclude'           => array(16, 20, 22,25, 27, 28, 30, 4), //* Enter ID's of parent categories to exclude from list
        'taxonomy'      => 'industrygroups',
        'parent'                   => '',
        );  

        $libcats=get_categories($libargs);  

        $i = 0;

        foreach($libcats as $lc){

            if( $i % 4 == 0 ) {
                /* INSERT CLEARFIX STUFF */
            }

            $i++;

            echo '<div class="col-lg-3 col-md-4 col-sm-6 col-xs-6 ">';

            $termlink = get_term_link( $lc->slug, 'industrygroups' );
        ?>


        <div class="thumbnail">
            <div class="caption">
                <br/><br/>   
                <h1><span class="label label-warning"><?php echo  $lc->count  ?></span></h1>
                <p> Symbols </p>

                <p> <a class="label label-default" href="https://wordpress.stackexchange.com/questions/158346/<?php echo $termlink; ?>"> View Group</a> </p> 

                <!-- basic text field display start -->
                <p>    <?php the_field('basictext', 'industrygroups_'.$lc->term_id); ?></p>
                <!-- basic text field display end -->

            </div>

            <!-- Get Image by Attachment ID Start-->
            <?php 

                $attachment_id = get_field('taximage', 'industrygroups_'.$lc->term_id);

                if ($attachment_id) {
                    $image = wp_get_attachment_image_src($attachment_id, 'industrygroup-img');

                    if ($image) {
                        ?>
                        <img class="img-responsive" src="<?php echo $image[0]; ?>" />
                        <?php 
                    } 
                } 

            ?>
            <!-- Get Image by Attachment ID End-->

        </div>

    <small><p class="text-center"> <a href="https://wordpress.stackexchange.com/questions/158346/<?php echo $termlink; ?>"> <?php echo $lc->name; ?></a> </p>  </small>

    <?php   echo '</div>'; }?>

</div>