Yes, you can do that.
Add a DataTrigger and bind to the respective property. Here's an example:
<DataTemplate.Triggers>
<DataTrigger Binding="{Binding Path=MyProperty}" Value="True">
<BeginStoryboard Storyboard="{StaticResource myStoryboard}"/>
</DataTrigger>
</DataTemplate.Triggers>
You can set the value to test for to anything you want. So you could set the storyboard to begin when your value changes to false. You can add as many DataTriggers (or other triggers) as you want.
Notice that in my example I reference a dummy property and storyboard.
When the property changes, the binding will be updated and will fire the trigger because of the databinding.
This technique should also work at startup.