I'm getting asked a similar question by two people at once. Here's an example of how an energy company tried to rip me off.
I was with nPower a few years ago and changed to another energy provider. On the day of changeover I took meter readings and supplied them to the new energy provider. In the UK the procedure IIRC is that the new energy provider gives those readings to ofgem (gov organisation overseeing energy companies), ofgem for whatever reason allows the energy company being left to pad/'estimate' the readings despite them being given meter readings to base their calculations on. nPower then padded the readings and claimed I owed them money. The argument then went back and forth for months despite being a simple one of "just use the actual meter readings, not your made-up ones", every excuse under the sun including trying to convince me that it's hardly worth using the real meter readings anyway, closing complaints I opened despite me telling them not to, until they eventually relented and admitted they owed me money.
It could have gone worse, I might have had to take them to the small claims court, and evidence is really bloody useful then.
@dank69 the only other question of yours that makes some kind of sense: "What? I highly doubt all of your different utilities are going to suddenly make all statements unavailable at the same time." - No, but my water company only produces 2 bi-annual statements so "statements less than three months old" wouldn't be useful there, and I think that phone company statements have been rejected for 'reasons'. The other issue can sometimes be "it needs to be your / your wife's name on the bill", and some companies only allow one name on the bill.
One other reason why I'd keep historical bills is that I run a business and HMRC requires 6 years worth of records, and since I claim a portion of my home expenses on my business, utility bills are relevant.