Chosen.js (chosen.css) and the bootstrap css both add CSS styles to your inputs (selects). Try to load chose.css after bootstrap.css:
  <link rel="stylesheet" href="bootstrap3/bootstrap-3.0.0-wip/dist/css/bootstrap.css">
  <link rel="stylesheet" href="docsupport/style.css">
  <link rel="stylesheet" href="docsupport/prism.css">
  <link rel="stylesheet" href="chosen.css">
  <style type="text/css" media="all">
    /* fix rtl for demo */
    .chosen-rtl .chosen-drop { left: -9000px; }
  </style>
After doing this see: Right border of the AddThis counter missing with Twitter's Bootstrap 3. It seems the CSS's universal selector to set box-sizing to border-box caused most of the trouble.
To fix this reset the box-sizing of the elements you apply chosen() on:
In the case of $('#select-input').chosen(); you will also set: 
#select-input
{
  -webkit-box-sizing: content-box;
     -moz-box-sizing: content-box;
      box-sizing: content-box;
}
NB by default chosen.js bundles an old version of jQuery. Twitter Bootstrap (javascript) requires the newest version (<2) of jQuery