How to invoke constants badly 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?
 A: There are periodically false proofs of the rapid decay (RD) Property (see Chatterji and Saloff-Coste - Introduction to the property of rapid decay) for cocompact lattices in semisimple Lie groups (Valette's conjecture). This is a functional-analytic property of these groups. As far as I understand, wrong proofs typically make use of some quantitative decreasing of coefficients, and this involves a "constant". The issue (sorry if I'm a bit imprecise) being that this "constant" actually depends on the dimension of something related to the induced representation restricted to a maximal compact subgroup.
A: It came to my mind what's perhaps the oldest example of this kind of mistake, so I add an answer to my own question: in 1821 Cauchy 'proved' that convergent sums of continuous functions are continuous, and later on Abel found counterexamples (see [1] for historical details). Of course Cauchy implicitly assumed uniform convergence, which means that he treated his $\delta$ as "more constant" than it was...
[1]: Sørensen, H. K. (2005). Exceptions and counterexamples: Understanding Abel's comment on Cauchy's Theorem. Historia Mathematica, 32(4), 453-480.
A: Edit: The original answer below refers to Nelson's attempt from 2011. Upon a cursory look at the afterword by Sam Buss and Terence Tao to Nelson's paper placed in arxiv in 2015 (after his death), it seems he later attempted to address the error referred to in the original answer below; it would be interesting to know what the experts think on how successful his efforts were or potentially can be.
Original Answer: Edward Nelson's recent project on finding inconsistency of arithmetic  (which was the subject of a MathOverflow Question) might be pertinent. The error, discovered by Terence Tao, seems to be the dependence of a constant on the underlying theory that Nelson did not account for.
A: A good example (of a somewhat different kind though) was given by Adian in the introduction to his book "The Burnside problem and identities in groups" (1975, English edition 1979), where he refutes the proof of the main result from his rival's book by stating that
``the conditions
$$
\begin{aligned}
&u_4 = u_1 +r_{25} \; \text{(p.145, line 10 from below)} \\
&r_{25} \ge u_{37} +54/e \; \text{(p.283, line 4 from below)} \\
&u_{37} >14\alpha +214/e, \; \text{where}\; \alpha=\varepsilon_{30}+u_{13} +6u_4 \; \text{(p.221, lines 11 and 12 from below)}
\end{aligned}
$$
give an obvious contradiction $u_4>r_{25}>u_{37}>u_4$.''
