It depends on the likelihood of the coincidence, and your method of identifying it. Feynman's example is nice: I was driving behind licence plate number XPF1034 today. What are the chances of THAT? In this case, you have to declare what you consider exceptional ahead of time, and if you don't, you have to consider how special your choice is. It's common sense, but it's not easy to formalize.
The level of coincidence is formalized by physicists as naturalness. You make a Baysian model which tells you how likely different possibilities are, and when the probability of a coincidence is too low, you look for a reason. But you make sure your level of coincidence is larger than one over the number of models you are searching for. If you are looking through a thousand places to find a one-in-a-thousand coincidence, you have found nothing.
For example, the mass of the electron is .5 MeV, while the mass of other things, quarks, and so on, are 1-4 GeV. This means the electron is too light by a factor of 1000. The leptons are generally lighter than their corresponding quark, but usually by a factor of 10 or so, so there is a 1% coincidence here. Is this significant? Possibly not. This is a borderline case. It would be nice to understand why the electron is so light, but it just might be a coincidence of our vacuum.
Another example is the Higgs mass, as compared to the Planck mass. The Higgs mass is around 1TeV, the Planck mass is 16 orders of magntiude bigger, so it's a coincidence of a million-billion. This is not something you can just shrug off.
For the cosmological constant, going in mass units, its 30 orders of magnitude (usually people raise to the fourth power, making a discrepancy of 120 orders of magnitude, but I prefer not to). This discrepancy is also a serious problem.
For other coincidences, it's Baysian common sense. Suppose you find that certain military drills simulating multiple hijackings are going on at the same time as 9/11. That's a weird coincidence, it gives you Baysian discomfort, like the electron mass, so you find it suspicious, but not terribly notable. Suppose you later figure out a precise set of drills can be used to stage 9/11, and it's not adjustible, it needs to be these drills and no others. Then when you see that the drills that were going on match these drills (consistently, up to classified details), and if some drills are missing, and then you find news reports that these too were going on, then it stops being like the electron mass, and starts being like the cosmological constant.
The formalizing of coincidence in Baysianism is used in less controversial ways every day. For example, if you extract viral DNA from a bacterial disease, and you keep extracting it, and you get more and more Baysian discomfort, at some pont you say "hold it, maybe this is really a viral disease." That's how you get new hypotheses. If they are correct, they will immediately straighten out all the Baysian discomfort.
But you have to be nimble, and use common sense, and know when you are fooling yourself, you know, the standard caveats.