Einstein's theory of special relativity does indeed state that objects with mass cannot reach or exceed the speed of light. This is known as the "cosmic speed limit." According to special relativity, as an object with mass accelerates, its energy and momentum increase, requiring more and more energy to continue accelerating. As the object approaches the speed of light, its energy and momentum would theoretically become infinite, which is not possible.
Regarding the idea that a massless object would gain mass as it travels at the speed of light, it is a common misconception. In special relativity, the mass of an object is not affected by its velocity. Instead, the theory introduces the concept of relativistic mass, which is a quantity that includes both an object's rest mass (its mass at rest) and its kinetic energy. As an object accelerates, its relativistic mass increases, but its rest mass remains constant.
The concept of relativistic mass can lead to confusion because it implies that an object's mass increases with velocity. However, it is important to note that this is a mathematical description and not an actual increase in the object's mass. In modern physics, the use of relativistic mass has fallen out of favor, and it is more common to consider the object's rest mass and its energy separately.
So, while the behavior of objects near the speed of light can be counterintuitive and different from classical Newtonian physics, it does not lead to a paradox. The theory of special relativity provides a consistent framework for understanding the behavior of objects with mass and light.