First, it's wise to keep a distinction between fiction and fact. Fiction is presented as entertainment. A diversion. It is not intended to be presented as real life or things that can occur in real life.
Time travel is possible only within the realms of the time dilation theory. As such, travel into the future is possible but not into the past.
Tony has a misconception that the speed obtained must be close to the speed of light. This is not true. The theory only states that the dilation rate would increase the faster one travels.
Other hypothetics such as wormholes have not been observed in nature and have not been produced in laboratory conditions. They probably do not exist.
In Albert Einstein's theories of relativity time dilation is manifested in two circumstances: Gravitational and Velocity. The effect is described by the Lorentz transformation.
In general relativity, clocks at lower potentials in a gravitational field — such as in close proximity to a planet — are found to be running slower. In special relativity, the time dilation effect is reciprocal: as observed from the point of view of any two clocks which are in motion with respect to each other, it will be the other party's clocks that is time dilated.
Thus the 'moving clock' is measured to be "running slow." The range of such variances in ordinary life, where v / c < 1 can be safely ignored. It is only when an object approaches speeds on the order of 30,000 km/s (1/10 of the speed of light) that time dilation becomes important.
Time dilation by the Lorentz factor was predicted by Joseph Larmor (1897). It has been tested a number of times. The routine work carried on in particle accelerators since the 1950s, such as those at CERN, is a continuously running test of the time dilation of special relativity. The specific experiments include:
Velocity time dilation tests:
Ives and Stilwell (1938, 1941), “An experimental study of the rate of a moving clock”, in two parts. The stated purpose of these experiments was to verify the time dilation effect, predicted by Lamor-Lorentz ether theory, due to motion through the ether using Einstein's suggestion that Doppler effect in canal rays would provide a suitable experiment. These experiments measured the Doppler shift of the radiation emitted from cathode rays, when viewed from directly in front and from directly behind. The high and low frequencies detected were not the classical values predicted.
Rossi and Hall (1941) compared the population of cosmic-ray produced muons at the top of a mountain to that observed at sea level. Although the travel time for the muons from the top of the mountain to the base is several muon half-lives, the muon sample at the base was only moderately reduced. This is explained by the time dilation attributed to their high speed relative to the experimenters. That is to say, the muons are decaying about 10 times slower than they would in a rest frame (that is, for "stationary observers").
Hasselkamp, Mondry, and Scharmann (1979) measured the Doppler shift from a source moving at right angles to the line of sight (the transverse Doppler shift) as deduced by Einstein (1905). Thus there is no transverse Doppler shift, and the lower frequency of the moving source can be attributed to the time dilation effect alone.
In contrast, gravitational time dilation (as treated in General Relativity) is not reciprocal: an observer at the top of a tower will observe that clocks at ground level tick slower, and observers on the ground will agree. Thus gravitational time dilation is agreed upon by all stationary observers, independent of their altitude.
Gravitational time dilation tests:
Pound, Rebka in 1959 measured the very slight gravitational red shift in the frequency of light emitted at a lower height, where Earth's gravitational field is relatively more intense. The results were within 10% of the predictions of general relativity. Later Pound and Snider (in 1964) derived an even closer result of 1%. This effect is as predicted by gravitational time dilation.
Velocity and gravitational time dilation combined-effect tests:
Hafele and Keating, in 1971, flew cesium atomic clocks east and west around the Earth in commercial airliners, to compare the elapsed time against that of a clock that remained at the US Naval Observatory. Two opposite effects came in to play. The clocks were expected to age more quickly (show a larger elapsed time) than the reference clock, since they were in a higher (weaker) gravitational potential for most of the trip. But also, contrastingly, the moving clocks were expected to age more slowly because of the speed of their travel. The gravitational effect was the larger, and the clocks suffered a net gain in elapsed time. The net gain was consistent with the difference between the predicted gravitational gain and the predicted velocity time loss. In 2005, the National Physical Laboratory in the United Kingdom reported their limited replication of this experiment. The NPL experiment differed from the original in that the cesium clocks were sent on a shorter trip (London-Washingon D. C. return), but the clocks were more accurate. The reported results are within 4% of the predictions of relativity.
The Global Positioning System can be considered a continuously operating experiment in both special and general relativity. The in-orbit clocks are corrected for both special and general relativistic time-dilation effects so they run at the same (average) rate as clocks at the surface of the Earth.
Time dilation and space flight:
Time dilation would make it possible for passengers in a fast moving vehicle to travel further into the future while aging very little, in that their great speed retards the rate of passage of onboard time. That is, the ship's clock (and according to relativity, any human travelling with it) shows less elapsed time than stationary clocks (such as on the home planet).
The answer regarding atomic particles in a time machine- it is true that these experiments have only been successful upon subatomic particles. I would not recommend the use of 'time machine' to describe this transfer of matter. It would appear to be more in the category of teleporting than in time travelling.