Time travel is real and it happens.
The clocks on the GPS satellites have to be calibrated to take into account that the clocks on some satellites will tick more slowly than others because the slower ticking clocks are traveling through time more slowly than others.
Time is relative, not a constant. The speed of light isn't even constant although, as Einstein pointed out, we always measure the speed of light as a constant. Our measurements are wrong because we will always measure the speed of light relative to our position in time. It's like two cars traveling down the highway at 100mph in the same direction in parallel lanes. Relative to each other, the two cars aren't moving fast at all. In fact, if both were traveling at exactly 100mph, each car would see the other as standing still. Though they are both traveling very fast, we can only measure the relative speed between the two. That's why our measurements aren't accurate.
Astronauts in orbit around the earth actually age more slowly than if they were on earth because the mass of earth accelerates time. The closer you are to a massive object, the faster time moves. This means we all travel through time at an infinitesimally different speed than every one else. People who live and work in skyscrapers move through time more slowly than those on the ground!
If we were to gather enough mass or displace enough mass, we could either accelerate or decelerate our speed through time to the point that it would effect time travel. This introduces a number of classical paradoxes, however it is still theoretically possible.
The other possibilities of time travel rely on worm holes and black hole event horizons. At the event horizon of black hole, essentially the edge of its effect, time and space cease to proceed. If you were to enter the event horizon, time would essentially cease completely while time outside the black hole proceeded. The problem is getting out of the event horizon and living through the encounter!
Wormholes are theorized to exist based upon the idea that time is a wave. If you imagine a ribbon accordianed together, you see that it undulates up and down. Time is that ribbon. We and everything else, travel along that ribbon. Now, the shortest distance between two points on that compressed ribbon aren't found along the strip, but between the peaks. Imagine two roads traveling through a deep gorge. One road winds down from the peak of one side of the gorge, through its bottom, then up the other side to the peak. The other road is using a bridge that simply spans the gorge in a straight line from the peak on one side to the other. The shortest time to travel between the two peaks is gained by using the bridge, not the road through the gorge. Wormholes work on that same principle, that time can be bridged.
With enough mass, or energy capable of displacing mass, time in one place can be manipulated to move faster or slower relative to time further away from the generated mass. So yes, time travel is possible and we do effect it to occur, and humans experience it always even if in such minute differences that we don't notice it.
Find a way, by manipulating mass and energy, to increase or decrease the effects of each relative to your current place in time, and you have what we conventionally think of as time travel.