Nature article on reproducibility in science.
According to that article, a (surprisingly) large number of experiments aren't reproducible, or at least there have been failed attempted reproductions. In one of the figures, it's said that 70% of scientists in physics & engineering have failed to reproduce someone else's results, and 50% have failed to reproduce their own.
Clearly, if something cannot be reproduced, its veracity is called into question. Also clearly, because there's only one particle accelerator with the power of the LHC in the world, we aren't able to independently reproduce LHC results. In fact, because 50% of physics & engineering experiments aren't reproducible by the original scientists, one might expect there's a 50% chance that if the people who originally built the LHC built another LHC, they will not reach the same results. How, then, do we know that the LHC results (such as the discovery of the Higgs boson) are robust? Or do we not know the LHC results are robust, and are effectively proceeding on faith that they are?
EDIT: As pointed out by Chris Hayes in the comments, I misinterpreted the Nature article. It says that 50% of physical scientists have failed to reproduce their own results, which is not the same statement as 50% of physics experiments aren't reproducible. This significantly eases the concern I had when I wrote the question. I'm leaving the question here however, because the core idea - how can we know the LHC's results are robust when we only have one LHC? - remains the same, and because innisfree wrote an excellent answer.