If you have a form which submits multiple values of the same form variable you can separate them as follows.
For i = 1 to Request.form("myforminput").Count
If Request.form("myforminput")(i) <> "" then
Response.write Request.form("myforminput")(i) & "<br />"
End if
Next
It might be worth trying the same approach - ie:
For i = 1 to (objUpload.Form("thefile").value).Count
If (objUpload.Form("thefile").value)(i) <> "" then
Response.write (objUpload.Form("thefile").value)(i) & "<br />"
End if
Next
Because there's a third party component involved it may complicate things so I can't guarantee this will work. Check the component's documentation if it is available, particularly with respect to counting the number of files uploaded.