HTML
<link rel="stylesheet" href="http://maxcdn.bootstrapcdn.com/bootstrap/2.2.2/css/bootstrap.min.css" />
<script type="text/javascript" src="http://ajax.googleapis.com/ajax/libs/jquery/1.9.1/jquery.min.js"></script>
<script type="text/javascript" src="http://maxcdn.bootstrapcdn.com/bootstrap/2.2.2/js/bootstrap.min.js"></script>
<link rel="stylesheet" href="http://cdn.rawgit.com/jdewit/bootstrap-timepicker/gh-pages/css/bootstrap-timepicker.min.css" />
<script type="text/javascript" src="http://cdn.rawgit.com/jdewit/bootstrap-timepicker/gh-pages/js/bootstrap-timepicker.js"></script>
<div class="input-append bootstrap-timepicker">
<asp:TextBox ID="txtTime" runat="server" CssClass="input-small" />
<span class="add-on"><i class="icon-time"></i></span>
<asp:Button Text="Submit" runat="server" OnClick="Submit" />
</div>
<script type="text/javascript">
$('[id*=txtTime]').timepicker();
</script>
Code
protected void Submit(object sender, EventArgs e)
{
DateTime time = DateTime.Parse(Request.Form[txtTime.UniqueID]);
}