Over the eons many have claimed that travelling backward in time is possible. Their main logic is that time can be stretched and distorted thanks to relativistic speed, why can it not be contracted or even reversed? To that end, millions of sophonts and even transapients have attempted to travel back in time with all sorts of contraptions, none of which as of yet can be proven to be successful.
On the opposite end of this spectrum, there have been many who have claimed to be time travellers from either the past or the future in various forms. Out of all the attempts, only a infinitesimally small fraction can be stated as inconclusive, while all others have proven to be a hoax or caused by some form of mental defect, dementia or memetics. Of the inconclusive claims, it is usually due to lack of evidence, or strange circumstances surrounding it that lend at least some credibility to the claims. No hard and fast results have come from study of these subjects or their stories.
As for proposed time machines, they have either disappeared, and assumed disintegrated, or failed to achieve any functionality at all. This includes forward projection while remaining in a stationary position. Although it is possible to "travel forward" in time, this has only been accomplished through high relativistic speed travel. All other claims have proven to be a hoax of some form or another.
A current theory on why time travel is possible but not provable is based in Quantum Theory that when an object of any mass travels back in time, to prevent causality paradoxes, the universe "splinters" a version of itself off in which the time traveller appears in, effectively sequestering the temporal anomaly. Of course this theory is totally unprovable as of current date, or if the Archai know, they aren't talking.
Sceptics of this Quantum Time Splinter Theory believe that because no time travellers have never come to this reality, that this cannot be true. Believers in this theory counter with the belief that because this may be the "core" reality, no time travellers can come here but can only spawn off into splinters of their splinter. So a time traveller in a splinter creates another version of reality forcing him self farther out into the unknown without ever realizing it. No matter what, all sides agree that no proof has yet been shown to back these ideas and they are only beliefs and conjectures.
The fascination of time travel and the prospect of affecting the past continues to entrance portions of society and until it is proven conclusively positively or negatively, it will probably continue to entrance sophonts for centuries to come, allowing this legend to persist.