You should probably rename your timers to keep them straight, something like SlideOpenTimer and SlideCloseTimer, etc. Would make it easier to understand what the timers are for.
I re-worked your timer events to hide the ListBox when the closing timer starts and to show the ListBox when the panel reaches its full width:
Private Sub Timer1_Tick(sender As Object, e As EventArgs) Handles Timer1.Tick
If Panel1.Width < 100 Then
Panel1.Width += 50
ElseIf Panel1.Width < 150 Then
Panel1.Width += 25
ElseIf Panel1.Width >= 150 Then
Timer1.Enabled = False
ListBox1.Visible = True
End If
End Sub
Private Sub Timer2_Tick(sender As Object, e As EventArgs) Handles Timer2.Tick
If ListBox1.Visible Then
ListBox1.Visible = False
End If
If Panel1.Width > 100 Then
Panel1.Width -= 50
ElseIf Panel1.Width > 25 Then
Panel1.Width -= 25
ElseIf Panel1.Width <= 25 Then
Timer2.Enabled = False
End If
End Sub