Everybody knows that Einstein's General Relativity predicted Mercury's anomalous motion, Black Holes, and other relativistic phenomena. But what specifically those predictions say to physicists? In this work, conflicts between hypothetical properties of the Black Hole concept in Einstein's General Relativity and in Modern Astrophysics are outlined as a platform for further studies of possible conceptual inconsistencies and paradoxes inside the theory. The arising issues are discussed in view of current Astrophysical observations of still mysterious galactic objects, Black Holes "swallowing" the surrounding matter, -- forever, but really?