In the original Rocky, Mickey tells Rocky that he is 76 years old:
Rocky, I'm seventy-six years old. Maybe you can be the winner I never was -- your shot is my last shot!
In Rocky III, Mickey dies, and we see his gravestone:
It says he was born in April, 1905 and died in August, 1981, meaning he only turned 76 four months earlier.
This wouldn't be such a problem if not for this tidbit, also from Rocky III, prior to Mickey's death:
ANNOUNCER: Another southpaw, champion Rocky Balboa, isn't worried about much these days. He's defended his title for the tenth time.
How could Rocky defend his title (which he won in Rocky II) ten times in four months? Even if Mickey was rounding his age up slightly in Rocky1, it would be difficult to pack the events of the first two movies, plus the 10 unseen title defenses between Rocky II and Rocky III into the time span available2.
As @steelerfan just pointed out in chat, Rocky Junior is a newborn in Rocky II, and appears to be 4-5 years old in Rocky III, reinforcing the idea that Mickey should have aged several years between I and III.
How much time passed between Rocky and Rocky III? Or in other words, how could Mickey remain 76 years old in the first, second, and third films?3
1 E.g., if he was really 75 years and 9 months old or whatever
2 Sticking with the hypothetical scenario mentioned in footnote 1, we would have to believe that Rocky met Mick, trained, fought Apollo Creed and lost (Rocky), then fought Creed again and won (Rocky II), then fought and won ten title defenses (between the end of II and the beginning of III), all in less than seven months.
3 In the interests of transparency and giving credit where credit is due, I confess that my cousin James noticed this and mentioned it on Facebook, after which I saw it and shamelessly stole it so I could ask it here.