I've had lots of small species of fish that lived far longer than their allotted span according to The All-Seeing And All-Knowing Eye of the internet. I'm talking here about species that are "supposed" to live 2 or 3 years but have survived for twice that long or more. And, yes, it's pretty obvious when one has finally begun to near its end; they slow down, lose their appetites, often grow thin, move clumsily and as little as possible. As long as they don't appear to be suffering unduly, I will let them wind down naturally, but if an old fish begins to appear weak and beyond hope, I will often end it quickly and humanely.
Big fish? I doubt very much I've had one die of old age; I think fish in general are capable of living much longer than we give them credit for. Big fish are old, and old fish are big; neither condition is likely to happen in a dinky little fish tank regardless of filtration, maintenance, water-changes, etc. Thinking back through my time in the hobby, I believe that most fish deaths I experienced, large or small, could be easily traced back to some mishap or accident that I could have prevented if I had foreseen it; catastrophic tank failures, electrical outages, accidental poisonings by airborne or other substances, poor choice of tankmates, incorrect diets, or combinations of these and many more factors likely kill most aquarium fish.
When one keels over dead without warning or without any immediately obvious cause, the common phrase is "died for no reason!" Yeah, sorry, everything has a reason, whether we can discern it...or admit it...or not. "Old age" is the most often blamed killer of those fish, and it might make us feel better, but...I don't think it happens very often at all.