In a nice and witty lecture titled "how to write mathematics badly" (available on YouTube at https://www.youtube.com/watch?v=ECQyFzzBHlo&t=23s), Jean-Pierre Serre describes various ways in which a paper can be poorly/confusingly/inaccurately written.
Around min 34:00 in the previous link, he criticizes the use of the word "constant", in particular in inequalities. The example he provides is of the type:
$$\|Af\|\le C\|f\|$$ for some constant $C$
where $A$ is a complicated operator depending on many parameters. In this case, he says, usually the only thing that the writer means is that $C$ does not depend on "some of the data" of the problem. He adds that this attitude "caused lots of mistakes".
What are examples of these mistakes? Has any significant piece of mathematics been rewritten or erased altogether because of some problem with proofs invoking "constants" too nonchalantly?