using hook_form_FORM_ID_alter with #states
Okay, I figured it out. Here is what I ended up with:

    function lodestar_form_user_profile_form_alter(&amp;$form, &amp;$form_state) {
    dpm($form);
    $form['field_your_nonprofit_connections']['und']['#states'] = array(
        'visible' =&gt; array(
            ':input[name="field_affiliation_type[und][Nonprofit]"]' =&gt; array('checked' =&gt; TRUE),
        ),
    );
}


This code got the element to toggle appropriately. Whew! I'm glad I finally solved it. Hopefully this question and answer will help someone else. It looks like the main thing I did wrong was having the incorrect ":input[name." The way I finally figured it out was by looking in the HTML itself via the element inspector. I should have tried that a long time ago.

Interestingly, the #states element never did show up in my dpm($form) output. I'm not sure why that is the case, but, apparently it doesn't need to show up there, because it is working fine without it.