Is this for real - a manager writing a letter to a third party just so that the employee knows she made a "mistake" when clearly almost everyone would see who's

Is this for real - a manager writing a letter to a third party just so that the employee knows she made a "mistake" when clearly almost everyone would see who's at fault? I can imagine that there would be such managers but I find it hard to imagine that someone would write a letter saying he/she did such a thing.