Twitter Bootstrap Tabs Upside Down

Use the following CSS to get what you expected. I added border to .tab-content class to get the content highlighted.

.tab-content {
padding:10px;
border:1px solid #ddd;
border-bottom:0px;
}
.nav-tabs {
border-bottom: 0px;
border-top: 1px solid #ddd;
}
.nav-tabs > li {
margin-bottom:0;
margin-top:-1px;
}

.nav-tabs > li > a {
padding-top: 8px;
padding-bottom: 8px;
line-height: 20px;
border: 1px solid transparent;
-moz-border-radius:0px;
-webkit-border-radius:0px;
border-radius:0px;
-moz-border-radius-bottomright: 5px;
-webkit-border-bottom-right-radius: 5px;
border-bottom-right-radius: 5px;
-moz-border-radius-bottomleft: 5px;
-webkit-border-bottom-left-radius: 5px;
border-bottom-left-radius: 5px;
}
.nav-tabs > .active > a, .nav-tabs > .active > a:hover, .nav-tabs > .active > a:focus {
color: #555555;
cursor: default;
background-color: #ffffff;
border: 1px solid #ddd;
border-top-color: transparent;
}

The HTML structure should be like this:

<div class="tabbable">
    <div class="tab-content">
        <div id="tab1" class="tab-pane active">
            tab1 content</div>
        <div id="tab2" class="tab-pane">
            tab2 content</div>
        <div id="tab3" class="tab-pane">
            tab3 content</div>
        <div id="tab4" class="tab-pane">
            tab4 content</div>
    </div>
    <ul class="nav nav-tabs">
        <li class="active"><a href="#tab1" data-toggle="tab">tab1</a></li>
        <li><a href="#tab2" data-toggle="tab">tab2</a></li>
        <li><a href="#tab3" data-toggle="tab">tab3</a></li>
        <li><a href="#tab4" data-toggle="tab">tab4</a></li>
    </ul>
</div>

Bootply example.

For those arriving looking for a solution for Bootstrap 4.

Bootstrap used to have tabs-below class to achieve this. But it was removed in version 3.

I "restored" this feature in the following code. You just need to add tabs-below class to your ul element. Check https://jsfiddle.net/oniltonmaciel/nhfrz2bc/ for the full solution.

css:

.tabs-below {
  border-bottom-width: 0px;
  border-top: 1px solid #dee2e6;
}

.tabs-below .nav-link {
    border: 1px solid transparent;
    border-top-left-radius: 0px;
    border-top-right-radius: 0px;
    border-bottom-left-radius: .25rem;
    border-bottom-right-radius: .25rem;
}

.tabs-below .nav-item {
  margin-bottom: 0px;
  margin-top: -1px;
}

.tabs-below .nav-item.show .nav-link, .tabs-below .nav-link.active {
  border-color: #fff #dee2e6 #dee2e6  #dee2e6;
}

html:

  <div class="tab-content" id="myTabContent">
    <div class="tab-pane fade show active" id="home" role="tabpanel" aria-labelledby="home-tab"><p>Contents of tab 1</p></div>
    <div class="tab-pane fade" id="profile" role="tabpanel" aria-labelledby="profile-tab"><p>Contents of tab 2</p></div>
    <div class="tab-pane fade" id="contact" role="tabpanel" aria-labelledby="contact-tab"><p>Contents of tab 3</p></div>
  </div>
  <ul class="nav nav-tabs tabs-below" id="myTab" role="tablist">
    <li class="nav-item">
      <a class="nav-link active" id="home-tab" data-toggle="tab" href="#home" role="tab" aria-controls="home" aria-selected="true">Home</a>
    </li>
    <li class="nav-item">
      <a class="nav-link" id="profile-tab" data-toggle="tab" href="#profile" role="tab" aria-controls="profile" aria-selected="false">Profile</a>
    </li>
    <li class="nav-item">
      <a class="nav-link" id="contact-tab" data-toggle="tab" href="#contact" role="tab" aria-controls="contact" aria-selected="false">Contact</a>
    </li>
  </ul>