Try this code:
<script language="javascript">
function sumquinto()
{
var Sumbog=0;
var val1= parseInt(document.getElementById("textbox1").value);
var val2= parseInt(document.getElementById("textbox2").value);
var val3= parseInt(document.getElementById("textbox3").value);
var val4= parseInt(document.getElementById("textbox4").value);
var val5= parseInt(document.getElementById("textbox5").value);
var val6= parseInt(document.getElementById("textbox6").value);
etc
if (val1 >=0)
{
Sumbog= Sumbog + val1
}
if (val2>=0)
{
Sumbog=Sumbog+val2
}
if (val3>=0)
{
Sumbog=Sumbog+val3
}
if (val4>=0)
{
Sumbog=Sumbog+val4
}
if (val5>=0)
{
Sumbog=Sumbog+val5
}
if (val6>=0)
{
etc
}
document.getElementById("txtTE2").value=Sumbog
}
</script>
<asp:TextBox ID="TextBox1" runat="server" onkeyup="sumquinto();"></asp:TextBox>
<asp:TextBox ID="TextBox2" runat="server" onkeyup="sumquinto();"></asp:TextBox>
<asp:TextBox ID="TextBox3" runat="server" onkeyup="sumquinto();"></asp:TextBox>
<asp:TextBox ID="TextBox4" runat="server" onkeyup="sumquinto();"></asp:TextBox>
and on your solution textbox dont add the onkeyup event.