Yes, age would have destroyed the bomb. The batteries would have gone bad within a decade, assuming modern tech, which may be out of place for Fallout. The pit doesn't degrade very fast, but some of the other materials used to boost performance will, such as tritium. (The "pit" is what we call the plutonium mass used for the Primary detonation.) I doubt the thing would have fired past 5 years.
Which is why you can't detonate the bomb on it's own. You need to use the pulse generator to cause the fissionable material to react, because the detonator that was built into the bomb does not work anymore.