To reproduce: use the following code:

RadDesktopAlert desktopAlert = new Telerik.WinControls.UI.RadDesktopAlert();

public Form1()
{
    InitializeComponent(); 
    
    desktopAlert.AutoClose = false;
    RadButtonElement radButtonElement = new RadButtonElement();
    radButtonElement.Click += RadDesktopAlertOkButtonClick;
    radButtonElement.Text = "OK";
    desktopAlert.ButtonItems.Add(radButtonElement); 
    desktopAlert.PopupAnimationDirection = RadDirection.Up;
    desktopAlert.PopupAnimationEasing = Telerik.WinControls.RadEasingType.InBack;

    BackgroundWorker bw = new BackgroundWorker();
    bw.DoWork += new DoWorkEventHandler(bw_DoWork);
    bw.RunWorkerAsync();
}

private void CreateAndShowAlert(string caption, string text)
{
    desktopAlert.CaptionText = caption;
    desktopAlert.ContentText = text;

    desktopAlert.Show();
}

void RadDesktopAlertOkButtonClick(object sender, EventArgs e)
{
    desktopAlert.Hide();
}

void bw_DoWork(object sender, DoWorkEventArgs e)
{
    int i = 0;
    while (true)
    {
        Thread.Sleep(1000);
        i++;
        MethodInvoker mi = new MethodInvoker(delegate() { this.CreateAndShowAlert("Alert " + i, "This is the " + i + "th alert"); });
        this.Invoke(mi);
    }
}

When I click the OK button, the DesktopAlert hides correctly, but after a few times (10 or more) the DesktopAlert stops appearing.

Workaround:

radDesktopAlert.Hide();
radDesktopAlert.Show();

or to switch off the animation:

radDesktopAlert.PopupAnimation = false;