Yes! But maybe not the way you're thinking.
We can (sort of) travel forward in time by going very fast. As you approach the speed of light, time passes significantly more slowly. Astronauts who spend six months on the ISS may feel like they have been up there for six months and one second, while on Earth, we have felt six months and two seconds. The effect grows as you increase your speed. If one goes 80% the speed of light, it might take six years to reach Alpha Centauri, while on Earth it will feel as if ten years have passed. At the speed of light, time does not pass. A photon may travel billions of light years, but it feels immediate for the photon.
The implications of this are that if you want to travel to the future, you would have to go very fast and then return to earth. That's impractical, however, and likely won't be achieved. We are not completely sure how all this works, but it's theorized by some that if we fully utilize the properties of wormholes (which we still have not discovered concrete evidence of), we may be able to travel forward and backward in time and space as we please.
Traveling forward in time is the non-issue here, philosophically, though the technology is many years off still. Traveling backward in time is the real problem for our brains. Take Barjavel's famous Grandfather Paradox:
If someone travels back in time and kills her grandfather when he is a child, then her grandfather never has her father, and therefore the traveller never could have been born to travel back in time to kill her grandfather.
This of course rings true with killing other family members paramount to your own birth, and also with "autoinfanticide", or, going back in time and killing yourself as a baby.
There are a few ways we get around this.
Novikov self-consistency principle, a.k.a., the Harry Potter model. Any action that you take after returning to the past was always the action that happened. For instance, take the scene in The Prisoner of Azkaban when they are all at Hagrid's place and a rock comes soaring through the window, causing them to see that the executioner is coming. Later on, Hermione and Harry go back in time and, seeing that they need to warn their past selves of the approaching executioner, Hermione throws a rock in the window. Therefore, every action she took in the past was always going to happen, and she fulfilled it by actually going back and doing it. This model is used heavily in Doctor Who.
Parallel universe theory #1: Nonexistence Theory, a.k.a., the It's A Wonderful Life model. You travel back in time, and upon returning to the future, you see the consequences of your actions. If you cause something that leads to your nonexistence (killing your grandfather is one such action), then return to the future, you will find yourself in a timeline where you never existed, though obviously you would still exist. It would be an alternate timeline where your life never happened. You existed in the past, and you exist now, but you did not exist in between.
Parallel universe theory #2: The Butterfly Effect. You go back in time and you accidentally step on a butterfly. The butterfly in the future was going to land on a boy's face and cause him immense joy. Since you killed that butterfly, the boy never experiences that joy, and grows up sad, then becomes an evil dictator. The Butterfly Effect theory states that by simply existing in the past, you are creating a new timeline. You may step on a microbe, or carry a virus to someone, or do some small action that has massive results. Often, massive things start with small actions. The Butterfly Effect is a realistic testament to the connectedness and importance of every single atom and every action of every being to the universal ecosystem.
The most likely answer is that when you travel back in time, you create a parallel universe. Many versions of multiverse theory state that there are a infinite number of universes, each with a different timeline. There is a universe for every "what-if" moment on earth, and every other planet in every galaxy. There's a universe where you're reading this on a train, and one where you're reading it in a park, and one where you're reading it and you have blue skin, and there are an infinite number of universes where everything on earth is exactly the same and things on other planets go differently. So if this multiverse theory is correct, and you travel back in time, is it possible that with every action you take, you are travelling between universes into other timelines?
Our understanding of time and space and multiverses is not quite there yet. We may never know the philosophical answer to time travel, even if we ever figure out the physical answer.