As the carrier of the gravitational interaction, graviton is considered to be spin-2 (tensor) boson, electrically uncharged, as well as massless since, according to General Relativity (GR), it travels along null geodesics at the speed of light (like photon). However, according to some alternative theories, gravity is propagated by a massive field, i.e. by a graviton with some small, nonzero mass. Ever since they were first introduced in 1939 by Fierz and Pauli, such so called theories of massive gravity have gained a significant attention due to their ability to provide a possible explanation for the accelerated expansion of the Universe without dark energy hypothesis, and due to important predictions that the velocity of gravitational waves (gravitons) should depend on their frequency, as well as that the effective gravitational potential should include a nonlinear (exponential) correction of Yukawa form, depending on the graviton Compton wavelength. Here we present a short overview of our investigations in which we considered a Yukawa-like modification of the Newtonian gravitational potential in the weak field approximation and its applications for obtaining the graviton mass bounds from the observed stellar orbits around the central supermassive black hole (SMBH) of our Galaxy. For that purpose we first derived the corresponding equations of motion which appeared to be highly nonlinear due to the Yukawa correction term in the potential, and used them to perform two-body simulations of the stellar orbits. The simulated orbits were then fitted to the observed orbit of S2 star around the Galactic Center in order to constrain the parameters of Yukawa gravity (the range of Yukawa interaction $\Lambda$ and universal constant $\delta$). It was found that the range of Yukawa interaction was on the order of several thousand astronomical units (AU), and assuming that this parameter corresponds to the Compton wavelength of graviton, we estimated the upper bound for graviton mass to $m_g < 2.9 \times 10^{-21} $ eV. This estimate was not only consistent with the LIGO estimate obtained from the first gravitational wave signal GW150914, but also it was obtained in an independent way. For that reason, it has been since 2019 included in the "Gauge and Higgs Boson Particle Listings" published by the Particle Data Group (PDG). We also discussed the possible influence of the bulk distribution of matter on this estimate, as well as the possibility for its improvement with future observations. We demonstrated that analysis of the observed stellar orbits around the Galactic Center in the frame of the massive gravity theories represent a very powerful tool for constraining the graviton mass and probing the GR predictions.