No, even if the temperature of a system were to increase indefinitely, the speed of atoms within that system would not exceed the speed of light. According to the theory of relativity, no particle with mass can attain or exceed the speed of light.
As the temperature of a system increases, the average speed of the atoms within it also increases. This is because temperature is a measure of the average kinetic energy of the particles in a system, and kinetic energy is related to the speed of particles. However, as the speed of particles approaches the speed of light, the increase in kinetic energy leads to an increase in mass according to Einstein's mass-energy equivalence equation (E=mc²). As mass increases, it requires more and more energy to further accelerate the particles, making it impossible to reach or exceed the speed of light.
Moreover, as particles with mass approach the speed of light, their relativistic mass increases, and it would take an infinite amount of energy to accelerate them to the speed of light. This is known as the mass-energy equivalence principle.
Therefore, no matter how high the temperature or the kinetic energy of the particles, they would never be able to reach or exceed the speed of light in a vacuum. The speed of light remains an upper limit on the speed of anything with mass in the universe, as far as our current understanding of physics goes.