I had to specify the UpdateSourceTrigger property. The Text property defaults to LostFocus, so when I changed to it PropertyChanged, it works as expected.

<TextBox x:Name="MultistageCountValue" 
  Text="{Binding MultistageCount, UpdateSourceTrigger=PropertyChanged}"
  PreviewTextInput="NumericTextBox_PreviewTextInput"
  TextChanged="MultistageCountValue_TextChanged"/>