Why does Application.Current == null in a WinForms application?

Well IMHO opinion the other SO answer is not really the way to go for windows forms, although maybe not incorrect.

Normally you would use ISynchronizeInvoke for such a feature in WinForms. Every container control implements this interface.

You'll need to BeginInvoke() method to marshall the call back to the proper thread.

Based on your previous question the code would become:

public class SomeObject : INotifyPropertyChanged
{
    private readonly ISynchronizeInvoke invoker;
    public SomeObject(ISynchronizeInvoke invoker)
    {
        this.invoker = invoker;
    }

    public decimal AlertLevel
    {
        get { return alertLevel; }
        set
        {
            if (alertLevel == value) return;
            alertLevel = value;
            OnPropertyChanged("AlertLevel");
        }
    }

    private void OnPropertyChanged(string propertyName)
    {
        if (PropertyChanged != null)
        {
            this.invoker.BeginInvoke((Action)(() =>
            PropertyChanged(this, new PropertyChangedEventArgs(propertyName))), null);

        }
    }
}

Where you pass the owning Form class to the constructor of SomeObject. The PropertyChanged will now raised on the UI thread of the owning form class.

Application.Current is Specific for WPF Application. Therefore when you are using WPF controls in WinForms Application you need to initialize instance of WPF Application. Do this in your WinForms Application.

if ( null == System.Windows.Application.Current )
{
   new System.Windows.Application();
}