Q: Ajax Tabcontainer Header cuts off in Chrome SOLVED. Problem below and as simple a fix as I can outline.
Visual Studio 2013 + AjaxToolkit 4.5.7.1213 (most recent stable).
Web forms app. I have a very simple page. It renders perfectly in IE compatibility view or Chrome's IE tab extension. However, in Chrome 33.0.1750.154 (couple weeks old version), or IE 11 the header of a tab in the tabcontainer is cut off horizontally; the bottom just isn't there. I have googled, and others had similar problems a while ago but nothing seems to easily resolve it. Doesn't matter if I use the inline HeaderText of the tab panel or a Header Text section, same thing happens.

I discovered that the problem resolved if I stopped using bootstrap.css.
Here is the problem:
Ajaxtoolkit has its own CSS, and this can be observed by looking at the source code for the tabcontainer online:
http://ajaxcontroltoolkit.codeplex.com/SourceControl/latest#Server/AjaxControlToolkit/Tabs/Tabs_resource.css
The default css class that the ajaxToolkit:TabContainer uses is .ajax__tab_tab. The offending code is in .ajax__tab_tab. It specifies a height of 13px. That cuts off the text. If I increase it to 18 px (on default font size), no issues.
Since the cssclass of ajaxToolkit:TabContainer can be specified, what I did was copy out most of the .ajax__tab_xp from the ajaxToolkit site, rename it to .customAjaxTab, place it in my own .css file, and then set CssClass="customAjaxTab" in my TabContainer.
Unfortunately, the images that the TabContainer default css calls use a "WebResource" call that I didn't get to work, so I made my images. My new css:
.customAjaxTab .ajax__tab_disabled { cursor: default; color: #A0A0A0; }
.customAjaxTab .ajax__tab_header {font-family:verdana,tahoma,helvetica;font-size:11px;background:url(Images/Tab_bottomline.png) repeat-x bottom;height:21px; }
.customAjaxTab .ajax__tab_header .ajax__tab_outer {padding-right:4px;background:url(Images/Tab_unselected.png) no-repeat right;height:21px;}
.customAjaxTab .ajax__tab_header .ajax__tab_inner {padding-left:3px;background:url(Images/Tab_unselected.png) no-repeat;}
.customAjaxTab .ajax__tab_header .ajax__tab_tab {height:18px;padding:4px;margin:0px;color:black;}
.customAjaxTab .ajax__tab_header .ajax__tab_hover .ajax__tab_outer {cursor:pointer;background:url(Images/Tab_unselected_hover.png) no-repeat right;}
.customAjaxTab .ajax__tab_header .ajax__tab_hover .ajax__tab_inner {cursor:pointer;background:url(Images/Tab_unselected_hover.png) no-repeat;}
.customAjaxTab .ajax__tab_header .ajax__tab_hover .ajax__tab_tab {cursor:pointer;}
.customAjaxTab .ajax__tab_header .ajax__tab_active .ajax__tab_outer {background:url(Images/Tab_selected.png) no-repeat right;}
.customAjaxTab .ajax__tab_header .ajax__tab_active .ajax__tab_inner {background:url(Images/Tab_selected.png) no-repeat;}
.customAjaxTab .ajax__tab_header .ajax__tab_active .ajax__tab_tab {color:black;}
.customAjaxTab .ajax__tab_body {font-family:verdana,tahoma,helvetica;font-size:10pt;border:1px solid #999999;border-top:0;padding:8px;background-color:#ffffff;}

This link has more on custom css for the TabContainer:
http://www.asp.net/ajaxlibrary/act_TabContainer.ashx

A: This worked for me, add in the bootstrap.css and bootstrap.min.css
.ajax__tab_tab
{
     -webkit-box-sizing: content-box!important;
     -moz-box-sizing: content-box!important;
     box-sizing: content-box!important;
}


A: xadriel's solution works - but, I have seen many comments stating 'Do not modify the Bootstrap.css file' - so I added xadriel's code segment to my site.css file and it works great! (Especially considering I'm using CDN linked CSS/JS so can't modify them...). 
.ajax__tab_tab
{
     -webkit-box-sizing: content-box!important;
     -moz-box-sizing: content-box!important;
     box-sizing: content-box!important;
}

I have an older VS2008/.NET 3.5 very large website that I am adding Bootstrap to, and I had to override a few Bootstrap settings by adding local site.css class names, then add those custom class names to the control needing fixing (this ensures changes only apply where needed and don't affect other areas). Some required !important to ensure they are changed (i.e. change color/background-color for a drop-down control where 'form-control' colors were being applied and I didn't want them). Doesn't seem necessary to add !important here as all I did was add this to site.css and it worked.
