It is real and very stressful. Be on your guard if you move house or live in a 'house of multiple occupation', or if you are a landlord. If you move write to all the banks etc. and keep records of all the replies to confirm this. Be aware that credit Cos are meant to check if someone is on the electoral roll, if they don't and you were not on the roll the Co likely takes the hit.
Simple stuff, really. I'm not convinced dumpster-diving is as widespread as suggested. I would still shred things, but bear in mind that even credit card statements do not contain the expiry date (or the code on the reverse side).