Please refer to the attached gif file demonstrating how to reproduce the problem.

Workaround: close the popup when pressing Backspace:

 this.radTextBox1.TextBoxElement.TextBoxItem.TextBoxControl.KeyDown+=TextBoxControl_KeyDown;
private void TextBoxControl_KeyDown(object sender, KeyEventArgs e)
{
    if (e.KeyData== Keys.Back)
    {
        TextBoxSpellChecker textBoxSpellChecker = _radSpellChecker.GetControlSpellChecker(typeof(RadTextBox)) as TextBoxSpellChecker;
        if (textBoxSpellChecker!=null)
        {
            textBoxSpellChecker.DropDownMenu.ClosePopup(RadPopupCloseReason.Keyboard);
        }
    }
}