A paradox called “grandmother paradox,” which is well known to the Special Relativity Theory and contemporary philosophy of time, is used to argue against the feasibility of travelling backward in time because of the logical and nomological problems such travel involves, and not only because we don’t have the technology to make it reality. We show that the same kind of problems would be encountered in leaping forward in time and then returning to the time of departure. A similar family of problems also arise in our having foreknowledge of the future without making any time travel. We point to the mysterious consequences of having access to a being, say a machine or a psychic, that can have infallible knowledge of the future and conveys this foreknowledge to human beings truthfully, without any lies or distortions. The cause of these mysterious consequences is the fact that such machines or psychics will raise logical and nomological complications reminiscent of the ones we encounter in time travel scenarios, and that is a strong reason why infallible foretellers cannot exist. We argue that we can have foreknowledge of the future, in principle, only within certain narrow limits, if at all.