bullant is absolutely right — you have to change those ids to classes.
That said, $(".bottom") is a jQuery object containing all elements with the class "bottom", and any method you call on that object will be executed on all the elements it contains.
What you really want to do is calling the method only on the one element that is a child of the clicked element, and that can be achieved with
PHP Code:
$('.clickme').click(function () {
$(this).find('.bottom').slideToggle('fast');
});