See also a more detailed scientific discussion of this subject
If fields are to have any effect on people, there must be a mechanism of interaction. The field must interact with the electric charges in the body. If the field is too low, that interaction will not produce any effect on the whole body; it will get lost in the noise. So we can compare the size of the effect produced by the field with the electrical noise present in the body already.
When we do this, the conclusion is that we have not identified any mechanism where fields below about 5 µT could produce an interaction that is big enough to rise above the noise and produce an effect on the whole body. We probably have to go to 50 µT or more for the field to be big enough to produce effects.
The level of field implicated in the epidemiology is less than 1 µT. So this suggests that there are no mechanisms operating at the level implicated in the epidemiology, and this is turn casts doubt on whether the associations found in the epidemiology can be real effects of fields or not.
Of course, this is not conclusive. Maybe there is a mechanism that can operate at these low levels, and we just haven't thought of it yet. Or maybe there's a clever way one of the mechanisms we have thought of can operate at lower levels than we realise.
But most scientists see the absence of an identified mechanism that could operate at the low levels of field involved in the epidemiology as one of the arguments against EMFs being a cause of cancer.
One particular aspect of this debate concerns the energy of fields - is it enough to break bonds and damage DNA?
Examples are sometimes given of where we have accepted that something causes disease without knowing the exact mechanism. Cholera being transmitted by water and smoking and lung cancer are two examples. But in both these cases, there is nothing inherently implausible about the mechanism (something in water and tobacco smoke), it's just that we didn't know the exact agent. So these aren't very good parallels with EMFs.