Solution 1:

I am late to this one by a while, but I'm going to blow your mind, ready?

The reason this is happening, is because you are calling bootstrap in, after you are calling jquery-ui in.

Literally, swap the two so that instead of:

<script src="http://code.jquery.com/ui/1.10.3/jquery-ui.js"></script>
<script src="js/bootstrap.min.js"></script>

it becomes

<script src="js/bootstrap.min.js"></script>
<script src="http://code.jquery.com/ui/1.10.3/jquery-ui.js"></script>

:)

Edit - 26/06/2015 - this keeps attracting interest months later so I thought it was worth an edit. I actually really like the noConflict solution offered in the comment underneath this answer and clarified by user Pretty Cool as a separate answer. As some have reported issues with the bootstrap tooltip when the scripts are swapped. I didn't experience that issue however because I downloaded jquery UI without the tooltip as I didn't need it because bootstrap. So this issue never came up for me.

Edit - 22/07/2015 - Don't confuse jquery-ui with jquery! While Bootstrap's JavaScript requires jQuery to be loaded before, it doesn't rely on jQuery-UI. So jquery-ui.js can be loaded after bootstrap.min.js, while jquery.js always needs to be loaded before Bootstrap.

Solution 2:

This is a comment on the top answer, but I felt it was worth its own answer because it helped me answer the problem.

If you want to keep Bootstrap declared after JQuery UI (I did because I wanted to use the Bootstrap tooltip), declaring the following (I declared it after $(document).ready) will allow the button to appear again (answer from https://stackoverflow.com/a/23428433/4660870)

var bootstrapButton = $.fn.button.noConflict() // return $.fn.button to previously assigned value
$.fn.bootstrapBtn = bootstrapButton            // give $().bootstrapBtn the Bootstrap functionality