Impossible? On the contrary, validated up the yin yang so to speak.
Here are a few sources on the experimental basis of SR (that's Special Relativity);
The Michelson and Gale Experiment (Nature 115 (1925), pg 566; Astrophys. J. 61 (1925), pg 137.)
g−2 Experiments as a Test of Special Relativity: Newman et al., Phys. Rev. Lett. 40 no. 21 (1978), pg 1355. P.S. Cooper et al., Physical Review Letters 42 (1979), pg 1386. Farley et al., Nuovo Cimento Vol 45, pg 281 (1966). Bailey et al., Nuovo Cimento 9A, pg 369 (1972). Bailey et al., Phys. Lett. 68B no. 2 (1977), pg 191.
The Brookhaven experiment to measure g−2 for muons, http://www.g-2.bnl.gov/
The Fizeau Experiment: Bilger et al., Phys. Rev. A5 (1972) pg 591. James and Sternberg, Nature 197 (1963), pg 1192.
Particle-Based Experiments: Nguyen, H.H., “CPT results from KTeV”, (2001). arXiv:hep-ex/0112046. Schwingenheuer, B. et al., “CPT tests in the neutral kaon system”, Phys. Rev. Lett., 74, pg 4376–4379, (1995). Carey, R.M. et al., “New Measurement of the Anomalous Magnetic Moment of the Positive Muon”, Phys. Rev. Lett., 82, pg 1632–1635, (1999). (Reinhardt's Ph.D. thesis, 2005)
Calorimetric Test of Special Relativity: .R. Walz, H.P. Noyes and R.L. Carezani, Physical Review A29 (1984), pg 2110.
Twin Paradox: C. Alley, “Proper Time Experiments in Gravitational Fields with Atomic Clocks, Aircraft, and Laser Light Pulses,” in Quantum Optics, Experimental Gravity, and Measurement Theory, eds. Pierre Meystre and Marlan O. Scully, Proceedings Conf. Bad Windsheim 1981, 1983 Plenum Press New York, ISBN 0-306-41354-X, pg 363–427. Bailey et al., “Measurements of relativistic time dilation for positive and negative muons in a circular orbit,” Nature 268 (July 28, 1977) pg 301.
Doppler Shift Measurements: McGowan et al., Phys. Rev. Lett. 70 no. 3 (1993), pg 251. Olin et al., Phys. Rev. D8 no. 6 (1973), pg 1633. Mandelberg and Witten, Journal Opt. Soc. Amer. 52, pg 529 (1962).
Measurements of Particle Lifetimes: D. Frisch and J. Smith, “Measurement of the Relativistic Time Dilation Using Mesons”, Am. J. Phys. 31 (1963) 342. Ayres et al., Phys. Rev. D3 no. 5 (1971), pg 1051.
The Ives and Stilwell Experiment: H.E. Ives and G.R. Stilwell, “An Experimental Study of the Rate of a Moving Atomic Clock”, J. Opt. Soc. Am. 28 pg 215–226 (1938); JOSA 31 pg 369–374 (1941).
Other Experiments: Coleman and Glashow, “Cosmic ray and Neutrino Tests of Special Relativity”, Coleman and Glashow, “High-Energy Tests of Lorentz Invariance”
The Trouton-Noble Experiment: Tomaschek, Ann. d Phys. 78 (1926), p743; 80 (1926), pg 509.
Zhang, "Special Relativity and its Experimental Foundations".
The Kennedy-Thorndike Experiment
Wolf and Petit, “Satellite test of special relativity using the global positioning system”, Phys. Rev. A 56, p4405 (1997).
http://math.ucr.edu/home/baez/physics/Relativity/SR/experiments.html
A couple of more articles:
*Relativity in the Global Positioning System
http://www.physorg.com/news9248.html