Is there a rule that says you cannot have 2 of the same ad format within the same page? The reason why I'm asking is I have 2 link ads (same format) on the same page and one of them is not serving ads.
You cannot have two link ads on the same page. See the program policies: https://www.google.com/adsense/policies The format of the ads doesn't matter. You can have the banner ads if you want.
If you mean link units then you're only allowed one. Apparently there's some built in safeguard on some sites that prevents more than the maximum number of ads being displayed - maybe that's why your second one has dropped off.