Ajax fired when Filefield "Add Another Item" button is clicked conflicts with my jquery
These tips are correct for Drupal 7.
Your custom code should be coordinated with Drupal AJAX Framework.
When you click "Add Another Item" this happens:  


AJAX handler sends all the form with all values, and with point to element which has triggered AJAX.
Server returns fully new field. For example, if there were 4 inputs in your field, server will return 5 inputs, not just last fifth.
After that new piece of HTML goes through all the behaviors. As you may know, behavior's attach method has two params: context and settings. context is just new HTML for your field.


So, if you add your JS to this schema correctly it must work.

Drupal.behaviors.YOURMODULE = {
  attach: function (context, settings) {
    $('.some-class', context).once('YOURMODULE', function() {
      // Do some work here
    });
  }
}