The statement you mentioned regarding objects with mass not being able to reach the speed of light is indeed a fundamental principle of special relativity, as proposed by Albert Einstein. According to this principle, as an object with mass approaches the speed of light, its relativistic mass increases, and it would require an infinite amount of energy to accelerate it to or beyond the speed of light.
Regarding the notion of a massless object gaining mass as it travels at the speed of light, it is important to clarify that in the framework of special relativity, mass is not considered to be an additive property. Instead, the concept of relativistic mass, which increases with velocity, was introduced in earlier formulations of relativity.
However, the concept of relativistic mass has fallen out of favor in modern physics. In contemporary understanding, it is more common to refer to the "rest mass" or "invariant mass" of an object, which remains constant regardless of its velocity. Massless particles, such as photons (particles of light), have zero rest mass.
The statement you mentioned does not lead to a paradox within the framework of special relativity. It simply highlights the behavior of objects with mass as they approach the speed of light and the fact that they cannot reach or exceed that speed.