EMF sensitivity is only "real" in the sense that it causes genuine suffering for those who are convinced they are suffering from it.
The topic isn't studied much because it's understood that the issue is not a physical matter. It's easy to demonstrate that someone's supposed EMF sensitivity matches to visual cues (seeing a WiFi router, seeing a cellular phone) but does not correlate to actual measured RF power.
Many people don't understand how electromagnetic radiation decays with distance. A WiFi router may emit 1 Watt of peak power during transmissions, but that power doesn't go directly into the body of someone nearby. If it did, there wouldn't be any power left for anyone to receive a signal on their phone, so this much is easy to prove. The amount of power reaching any one person minus the amount of power reflected away from their body would be measured in picowatts or less. A picowatt is 0.000000000001 Watts, which isn't enough power to do much of anything.
A common mistake is to confuse studies at different frequencies or power levels, or to confuse electromagnetic radiation with magnetic fields. For example,
@Pyrrhus posted a study above showing that extremely powerful pulses at 50Hz and 1mT could cause observable effects in slices of animal brains. Such a study might be useful in the context of Transcranial Magnetic Stimulation (TMS), but it's not relevant to something like using your cellular phone due to the different frequencies (differing by a factor of 100,000,000) and different types of field (electromagnetic waves compared to magnetic pulses).
To put it in perspective: The field strength used in that study was 1 mT. A typical refrigerator magnet, which can hold papers to a steel surface, is around 5mT. Your cellular phone and WiFi router certainly aren't coming anywhere near that strength in the B-field.
I could write all day about this (it overlaps with my academic background) but the unfortunate reality is that it probably doesn't matter much. Those who are convinced that they react to RF radiation sickness are not actually suffering from direct effects of electromagnetic radiation in the air. The types of reactions experienced by sufferers don't actually correlate to measured RF levels, but rather to the perception of the presence of RF generating devices.