Multiple File Uploads fail when webform is placed in a block
Turns out that I had, due to custom markup added to the webform, nested &lt;form&gt; tags. Per the spec, this is a big no-no. What happens in the older browsers appeared to have been that the form AJAX submission tried to use the form ID of the most recent &lt;form&gt; tag, which, in my case, was empty. It triggered this in file.module:

if (empty($_POST['form_build_id']) || $form_build_id != $_POST['form_build_id']) {
  // Invalid request.
  drupal_set_message(t('An unrecoverable error occurred. The uploaded file likely exceeded the maximum file size (@size) that this server supports.', 
    array('@size' =&gt; format_size(file_upload_max_size()))), 'error');
  ...
  ...


Removing the unneeded &lt;form&gt; tags resolved the issue.