It's when those institutions lose credibility that things fall apart. If CERN came out with a press release tomorrow that said that actually the theory of relativity was wrong and the universe operates by newtonian mechanics, I would not only doubt that press release, but also everything else CERN put out recently. If a certain group started calling anyone who asked questions like "well how are GPS satellites so accurate" and "why does Mercury's orbit precess instead of being a perfect ellipse" was called a "science denier", well I would start doubting the credibility of that group.