It should be noted that Petr Vopenka himself did not believe in the principle! Here is the story, taken from Adamek and Rosicky Locally Presentable and Accessible Categories (p. 278-279).

The story of Vopenka's principle (as related to the authors by Petr Vopenka) is that of a practical joke which misfired: In the 1960's P. Vopenka was repelled by the multitude of large cardinals which emerged in set theory. When he constructed, in collaboration with Z. Hedrlin and A. Pultr, a rigid graph on every set (see Lemma 2.64), he came to the conclusion that, with some more effort, a large rigid class of graphs must surely be also constructible. He then decided to tease set-theorists: he introduced a new principle (known today as Vopenka's principle), and proved some consequences concerning large cardinals. He hoped that some set-theorists would continue this line of research (which they did) until somebody showed that the principle was nonsense. However the latter never materialized — after a number of unsuccessful attempts at constructing a large rigid class of graphs, Vopenka's principle received its name from Vopenka's disciples. One of them, T. J. Jech, made Vopenka's principle widely known.

It should be noted that Petr Vopěnka himself did not believe in the principle! Here is the story, taken from Adámek and Rosický Locally Presentable and Accessible Categories (p. 278-279).

The story of Vopěnka's principle (as related to the authors by Petr Vopěnka) is that of a practical joke which misfired: In the 1960's P. Vopěnka was repelled by the multitude of large cardinals which emerged in set theory. When he constructed, in collaboration with Z. Hedrlín and A. Pultr, a rigid graph on every set (see Lemma 2.64), he came to the conclusion that, with some more effort, a large rigid class of graphs must surely be also constructible. He then decided to tease set-theorists: he introduced a new principle (known today as Vopěnka's principle), and proved some consequences concerning large cardinals. He hoped that some set-theorists would continue this line of research (which they did) until somebody showed that the principle was nonsense. However the latter never materialized — after a number of unsuccessful attempts at constructing a large rigid class of graphs, Vopěnka's principle received its name from Vopěnka's disciples. One of them, T. J. Jech, made Vopěnka's principle widely known.