And I'm not saying they're *impossible*, it's just that they're getting reallllly hard to maintain in the modern age. I mean sure, if you dress up in a costume and beat up some criminals, and are super-careful, you probably won't get found out. But if you start messing around with powerful people or the government, chances are you're gonna start leaving your DNA around, or get caught on some satellites leaving your apartment or... let's just say it's not tenable in the long run.
Now, if you've got powers, you can keep the secret a lot longer. The Flashes have a HUGE advantage here, as they can zip away from their secret identities without being noticed by cameras. As do characters who shrink or turn invisible and whatnot. But regular guys n gals? If they get on the government's radar, it's just a matter of time. Unless, of course, they're a billionaire - Bruce Wayne probably drops at least ten million bucks a year keeping the hounds at bay.