Lets play a game. Do you think it's possible to un-check "Emergency" on the settings form, and save that change?

honeybadger.module

<?php

/**
 * Form builder for settings form.
 */
function honeybadger_settings_form($form, &$form_state) {
  $form['honeybadger_api_key'] = array(
    '#type' => 'textfield',
    '#title' => t('Honeybadger.io API key'),
    '#default_value' => variable_get('honeybadger_api_key'. ''),
    '#description' => t('Enter the <strong>Private API Key</strong> from the Setttings section of the Honeybadger.io project that corresponds with this site.')
  );

  $form['honeybadger_api_levels'] = array(
    '#type' => 'checkboxes',
    '#title' => t('Error levels to be sent to Honeybadger.io'),
    '#description' => t('Only errors that correspond with the checked error levels will be sent to Honeybadger.io.'),
    '#options' => array(
        WATCHDOG_EMERGENCY => t('Emergency'),
        WATCHDOG_ALERT => t('Alert'),
        WATCHDOG_CRITICAL => t('Critical'),
        WATCHDOG_ERROR => t('Error'),
        WATCHDOG_WARNING => t('Warning'),
        WATCHDOG_NOTICE => t('Notice'),
        WATCHDOG_INFO => t('Info'),
        WATCHDOG_DEBUG => t('Debug'),
      ),
    '#default_value' => variable_get('honeybadger_api_levels', array()),
  );

  return system_settings_form($form);
}