The chances that a false 3-sigma excess is observed in LHC experiments is so high that it happens of order monthly, and indeed these excesses are not usually reported or pointed out as anything of note.
The only thing that can be estimated with some certainty is the look elsewhere effect within a given, well-defined analytical problem. Indeed Monte Carlo simulations are the method of choice. For example, if you have a number of bins in your binned data, you can evaluate how that changes the true (global) probability of observing a 3-sigma excess in any one such bin. However, the truth is much more complicated, because in addition to various bins in a given analysis, there are also different parameters in the same analysis which contribute too. Then not everything would count as an excess because you may have requirements, e.g. from the resolution of your detector, that neighbouring bins show a consistent excess. It gets difficult quickly to model accurately. But then there are also many different kinds of analyses, which each also contribute to the look elsewhere effect that is relevant when assessing something from the news. This is essentially impossible to quantify.
At the end of the day, if you include every plot made by every of the thousands of graduate students on the LHC experiments you can easily see that the true look elsewhere effect is completely unquantifiable. For illustration, the probability of a five-sigma excess is somewhere in the $10^{-7}$ ballpark by definition. With, say, 10,000 analysts working on LHC experiments, each making just one new plot a week, and the experiment running for a decade, that's already close to this trial factor. Based on this simple estimate one could expect even a five-sigma excess to appear in an LHC analysis purely as a statistical fluke.
So to answer your question: It is virtually guaranteed that multiple false three sigma excesses will continue to pop up in LHC data.
