We all know that smoking had been around for a long time before we discovered it causes lung cancer and other diseases on a massive scale. Could EMFs from the power system be similar - is there any risk at all of discovering they pose a really major health problem only after we've been exposed to them for a long time?
The answer is that this almost certainly couldn't happen. We have looked for links between EMFs and many different diseases without finding anything clearcut or dramatic. The debate over EMFs is mainly about whether magnetic fields might be a cause of just a few percent of a disease, childhood leukaemia, which is fairly rare to start with. If there was anything as major as the link between smoking and lung cancer, we'd almost certainly have found it before now.
Magnetic fields have been around for over a century - see here for what we can learn by looking at how they've increased over that time. The radiofrequency EMFs from e.g. mobile phones and other even newer technologies have been around for much shorter periods and there's a debate about whether or not there could be a major health risk from them that hasn't emerged yet. But this site covers power-frequency EMFs only - see here for some links for radiofrequencies.
Smoking is sometimes cited as an example of how we can accept a causal link on epidemiological evidence alone without knowing the mechanism (cholera is another example). But there's nothing implausible about the chemicals in tobacco smoke causing cancer even if we didn't know exactly which chemical it was, so this isn't a good parallel to magnetic fields where there are arguments that there cannot be a plausible mechanism because the fields are too weak.