I had forgotten about the 2005-06 email correspondence reproduced below, but my collaborator Xavier Calmet reminded me of it today and I was able to find these messages.
The idea of a minimal length of order the Planck length, arising due to quantum gravity (i.e., quantum fluctuations in the structure of spacetime), is now widely accepted by theoretical physicists. But as Professor Mead (University of Minnesota, now retired) elaborates, based on his own experience, it was considered preposterous for a long time.
Large groups of people can be wrong for long periods of time -- in financial markets, academia, even theoretical physics.
Our paper, referred to by Mead, is
Minimum Length from Quantum Mechanics and Classical General Relativity
X. Calmet, M. Graesser, and S. Hsu
Phys Rev Letters Vol. 93, 21101 (2004)
The related idea, first formulated by R. Buniy, A. Zee, and myself, that the structure of Hilbert Space itself is likely discrete (or "granular") at some fundamental level, is currently considered preposterous, but time will tell.
At bottom I include a relevant excerpt from correspondence with Freeman Dyson in 2005.
Dear Drs. Calmet, Graesser, Hsu,
I read with interest your article in Phys Rev Letters Vol. 93, 21101 (2004), and was pleasantly surprised to see my 1964 paper cited (second citation of your ref. 1). Not many people have cited this paper, and I think it was pretty much forgotten the day it was published, & has remained so ever since. To me, your paper shows again that, no matter how one looks at it, one runs into problems trying to measure a distance (or synchronize clocks) with greater accuracy than the Planck length (or time).
I feel rather gratified that the physics community, which back then considered the idea of the Planck length as a fundamental limitation to be quite preposterous, has since come around to (more or less) my opinion. Obviously, I deserve ZERO credit for this, since I'm sure that the people who finally reached this conclusion, whoever they were, were unaware of my work. To me, this is better than if they had been influenced by me, since it's good to know that the principles of physics lead to this conclusion, rather than the influence of an individual. I hope that makes sense. ...
You might be amused by one story about how I finally got the (first) paper published after 5 years of referee problems. A whole series of referees had claimed that my eq. (1), which is related to your eq. (1), could not be true. I suspect that they just didn't want to read any further. Nothing I could say would convince them, though I'm sure you would agree that the result is transparently obvious. So I submitted another paper which consisted of nothing but a lengthy detailed proof of eq. (1), without mentioning the connection with the gravitation paper. The referees of THAT paper rejected it on the grounds that the result was trivially obvious!! When I pointed out this discrepancy to the editors, I got the gravitation paper reconsidered and eventually published.
But back then no one considered the Planck length to be a candidate as a fundamental limitation. Well, almost no one. I did receive support from Henry Primakoff, David Bohm, and Roger Penrose. As far as I can recall, these were the only theoretical physicists of note who were willing to take this idea seriously (and I talked to many, in addition to reading the reports of all the referees).
Well anyway, I greet you, thank you for your paper and for the citation, and hope you haven't found this e-mail too boring.
C. Alden Mead
Dear Dr. Mead,
Thank you very much for your email message. It is fascinating to learn the history behind your work. We found your paper to be clearly written and useful.
Amusingly, we state at the beginning of our paper something like "it is widely believed..." that there is a fundamental Planck-length limit. I am sure your paper made a contribution to this change in attitude. The paper is not obscure as we were able to find it without much digging.
Your story about the vicissitudes of publishing rings true to me. I find such stories reassuring given the annoying obstacles we all face in trying to make our little contributions to science.
Finally, we intend to have a look at your second paper. Perhaps we will find another interesting application of your ideas.
Xavier CalmetMichael Graesser
Many thanks for your kind reply. I find the information quite interesting, though as you say it leaves some historical questions unanswered. I think that Planck himself arrived at his length by purely dimensional considerations, and he supposedly considered this very important.
As you point out, it's physically very reasonable, perhaps more so in view of more recent developments. It seemed physically reasonable to me back in 1959, but not to most of the mainstream theorists of the time.
I think that physical considerations (such as yours and mine) and mathematical ones should support and complement each other. The Heisenberg-Bohr thought experiments tell us what a correct mathematical formalism should provide, and the formal quantum mechanics does this and, of course, much more. Same with the principle of equivalence and general relativity. Now, the physical ideas regarding the Planck length & time may serve as a guide in constructing a satisfactory formalism. Perhaps string theory will prove to be the answer, but I must admit that I'm ignorant of all details of that theory.
Anyway, I'm delighted to correspond with all of you as much as you wish, but I emphasize that I don't want to be intrusive or become a nuisance.
As my wife has written you (her idea, not mine), your e-mail was a nice birthday present.
Kindest Regards, Alden
... to me the most interesting is the discrete Hilbert Space paper, especially your reference  proving that lengths cannot be measured with error smaller than the Planck length. I was unaware of this reference but I had reached the same conclusion independently.