WebClient asynchronous downloader not working properly

First of all, the main reason your progress bar isn't working is because of this:

Dim percentage As Integer = (CType((total_dl / total_dl_size), Integer) * 100)

The code will first evaluate total_dl / total_dl_size, say it results in 0.34, then it will convert that into an integer which will result in 0 (0.34 is rounded down to zero, since integers have no decimals), and finally it multiplies that 0 with 100 (which still results in 0).
What you've got to do is to multiply the dividend with 100 first so that the result will go from 0-100 instead of 0-1: (total_dl * 100) / total_dl_size.

As for thread-safety (invoking) I always use this extension method that I created:

Imports System.Runtime.CompilerServices

Public Module Extensions
    <Extension()> _
    Public Sub InvokeIfRequired(ByVal Control As Control, ByVal Method As [Delegate], ByVal ParamArray Parameters As Object())
        If Parameters Is Nothing OrElse _
            Parameters.Length = 0 Then Parameters = Nothing 'If Parameters is null or has a length of zero then no parameters should be passed.
        If Control.InvokeRequired = True Then
            Control.Invoke(Method, Parameters)
        Else
            Method.DynamicInvoke(Parameters)
        End If
    End Sub
End Module

(preferrably put it in another file)

That, together with Lambda expressions (introduced in Visual Studio 2010), will greatly simplify invocation for you. This is because instead of putting the If InvokeRequired pattern everywhere:

If Me.InvokeRequired Then
    Me.Invoke(New Action(AddressOf SomeMethod), params)
Else
    SomeMethod()
End If

you only need to type:

Me.InvokeIfRequired(AddressOf SomeMethod, params)

and the extension method will do the rest for you.

And if you use lambda expressions you can create methods dynamically:

Me.InvokeIfRequired(Sub()
                        Label1.Text = "Hello world!"
                        ProgressBar1.Value += 1
                    End Sub)

Now, to your code.

I have separated your code a bit more so that it's easier to deal with. For starters, instead of copy-pasting the download code to the DownloadFileCompleted event handler I made a more generic method called DownloadFile().

''' <summary>
''' Downloads a file from the specified URL with the specified credentials.
''' </summary>
''' <param name="URL">The URL of the file.</param>
''' <param name="Username">The username which to login with.</param>
''' <param name="Password">The password which to login with.</param>
''' <remarks></remarks>
Private Sub DownloadFile(ByVal URL As String, ByVal Username As String, ByVal Password As String)
    If wc.IsBusy = True Then Throw New Exception("A download is already ongoing!")

    wc.Credentials = New NetworkCredential(dl_user, dl_pass)
    total_dl_size = GetDownloadSize(URL, Username, Password)

    Try
        Dim FileName As String = Path.GetFileName(URL)
        AppendWarning("Downloading " & FileName & "...")
        wc.DownloadFileAsync(New Uri(URL), Path.Combine(Environment.GetFolderPath(Environment.SpecialFolder.Desktop), FileName))
    Catch ex As Exception
        AppendWarning("-ERR: Could not download file: " & Path.GetFileName(URL))
    End Try
End Sub

As you see I also made a generic method for outputting warnings and error messages:

''' <summary>
''' (Thread-safe) Appends a warning or status message to the "tb_warnings" text box.
''' </summary>
''' <param name="Text">The text to append.</param>
''' <remarks></remarks>
Private Sub AppendWarning(ByVal Text As String)
    Me.InvokeIfRequired(Sub() tb_warnings.AppendText(Text & Environment.NewLine))
End Sub

Here's the entire code, it works properly for me:

Private dl_user As String = "someusername"
Private dl_pass As String = "somepassword"

Private dl_urls As String() = {"URL1", "URL2"} 'Temporary. Use your own code.
Private total_dl_size As Long = 0
Private total_dl As Long = 0

Dim WithEvents wc As New System.Net.WebClient()
Dim downloads As New Queue(Of String)

Private Sub Form1_Load(sender As System.Object, e As System.EventArgs) Handles MyBase.Load
    'Populate the download queue.
    downloads.Enqueue(dl_urls(0)) 'Temporary. Use your own code here.
    downloads.Enqueue(dl_urls(1))
End Sub

'The download button.
Private Sub Button1_Click(sender As System.Object, e As System.EventArgs) Handles Button1.Click
    'Do your pre-download stuff here.

    DownloadFile(downloads.Dequeue(), dl_user, dl_pass) 'Download the first file.
End Sub

''' <summary>
''' Downloads a file from the specified URL with the specified credentials.
''' </summary>
''' <param name="URL">The URL of the file.</param>
''' <param name="Username">The username which to login with.</param>
''' <param name="Password">The password which to login with.</param>
''' <remarks></remarks>
Private Sub DownloadFile(ByVal URL As String, ByVal Username As String, ByVal Password As String)
    If wc.IsBusy = True Then Throw New Exception("A download is already ongoing!")

    wc.Credentials = New NetworkCredential(dl_user, dl_pass) 'Set the credentials.
    total_dl_size = GetDownloadSize(URL, Username, Password) 'Get the size of the current file.

    Try
        Dim FileName As String = Path.GetFileName(URL) 'Get the current file's name.
        AppendWarning("Downloading " & FileName & "...") 'Download notice.
        wc.DownloadFileAsync(New Uri(URL), Path.Combine(Environment.GetFolderPath(Environment.SpecialFolder.Desktop), FileName)) 'Download the file to the desktop (use your own path here).
    Catch ex As Exception
        AppendWarning("-ERR: Could not download file: " & Path.GetFileName(URL))
    End Try
End Sub

''' <summary>
''' (Thread-safe) Appends a warning or status message to the "tb_warnings" text box.
''' </summary>
''' <param name="Text">The text to append.</param>
''' <remarks></remarks>
Private Sub AppendWarning(ByVal Text As String)
    Me.InvokeIfRequired(Sub() tb_warnings.AppendText(Text & Environment.NewLine))
End Sub

Private Sub wc_DownloadProgressChanged(sender As Object, e As System.Net.DownloadProgressChangedEventArgs) Handles wc.DownloadProgressChanged
    Me.InvokeIfRequired(Sub()
                            Dim Progress As Integer = CType(Math.Round((e.BytesReceived * 100) / total_dl_size), Integer)
                            If Progress > 100 Then Progress = 100
                            If Progress < 0 Then Progress = 0
                            ProgressBar1.Value = Progress
                        End Sub)
End Sub

Private Sub wc_DownloadFileCompleted(sender As Object, e As System.ComponentModel.AsyncCompletedEventArgs) Handles wc.DownloadFileCompleted
    If e.Cancelled Then
        MessageBox.Show(e.Cancelled)

    ElseIf Not e.Error Is Nothing Then
        MessageBox.Show(e.Error.Message)

    Else
        If downloads.Count > 0 Then
            DownloadFile(downloads.Dequeue(), dl_user, dl_pass) 'Download the next file.
        Else
            AppendWarning("Download complete!")
        End If

    End If
End Sub

Some other things to keep in mind:

• The MsgBox() function exists purely for backwards compatibility. You should use .NET's standard MessageBox.Show() method instead.

• String concatenation should be done using the ampersand (&) instead of the plus (+). See why.

• Concatenating paths should always be done using Path.Combine() as it will ensure to create a proper path. If you input anything invalid it'll throw an exception.

  Usage:

  Path.Combine(Path1, Path2, Path3, ...)
  Path.Combine("C:\", "Foo") 'Results in: C:\Foo
  Path.Combine("C:\", "Foo", "Bar", "Hello World.txt") 'Results in: C:\Foo\Bar\Hello World.txt
  

Hope this helps!