It should be noted that Petr Vopenka Vopěnka himself did not believe in the principle! Here is the story, taken from Adamek Adámek and Rosicky Rosický Locally Presentable and Accessible Categories (p. 278-279).
The story of Vopenka's Vopěnka's principle (as related to the authors by Petr VopenkaVopěnka) is that of a practical joke which misfired: In the 1960's P. Vopenka Vopěnka was repelled by the multitude of large cardinals which emerged in set theory. When he constructed, in collaboration with Z. Hedrlin 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 Vopenka's 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, Vopenka's Vopěnka's principle received its name from Vopenka's Vopěnka's disciples. One of them, T. J. Jech, made Vopenka's Vopěnka's principle widely known.