Sometimes fallibilists argue that math cannot have certainty because performing a proof is a physical process, and during physical processes things can go wrong (e.g. i could be drugged to unconsciousness and then awake with tampered memories such that I thought I'd completely the proof correctly when I hadn't). This argument is correct, but it is only an argument for the first, lesser meaning of fallibility. Although it gives an example demonstrating the possibility of a mistake, it does not show that mistakes are common.
A similar kind of argument is made by fallibilists with inductivists. We may point out that, as a matter of logic, inductive conclusions do not deductively follow from their premises, and therefore they are fallible. Again, this is an argument for fallibility in the first sense -- error is possible -- but it does not say whether error is common or not.
One result of this situation is that some people are converted to fallibilism but only in the first sense. When they encounter people who embrace fallibilism in the deeper sense, they become confused because these people discuss fallibilism but in a different way than they understand it. There can be further confusion because both groups identify themselves by the same label, "fallibilists", and may then wonder why they are disagreeing so much.
The more thorough meaning of fallibilism is required for most important fallibilist arguments. This is known to many anti-fallibilists who claim fallibilism is stupid and useless because not a lot of interesting truths follow from it (they have in mind the more limited meaning of fallibilism). And emphasizing that error is possible could be deemed misleading if it is in fact very very rare and perhaps even negligible.
Here are some examples of how the stronger meaning of fallibilism leads to important conclusions the weaker meaning does not:
Should parents take seriously the possibility that, in the face of a disagreement, their child might be in the right? If mistakes are common, including mistakes by parents, then yes they should. This is a clear implication from the strong meaning of fallibilism. But on the other hand if the parent having made a mistake is only a very remote possibility, one in a million, then one could considering taking a different attitude.
Should lovers who think they won't end up with broken hearts take seriously the possibility that their knowledge of how to avoid being hurt may contain a mistake? That depends if mistakes are commonplace or extremely rare. If the rate of making mistakes like that is one per hundred million couples then it's not worth worrying about. If it's one per two couples then it'd be crazy not to think about it a lot.
When a person seems to misunderstand my argument, should I believe he is doing it deliberately (perhaps because he sees that it refutes his position)? If mistakes in understanding arguments are extremely rare, then it would follow that it's usually deliberate. But if mistakes are common, then I shouldn't take it to be deliberate.
In general, when I disagree with someone, is he mistaken, am I mistaken, or is he a bad person? If mistakes are common, either of us could be mistaken. If mistakes are extraordinary rare, then I may have to conclude he is a bad person who wants to adopt mistaken ideas due to bias or some other factor. This is especially true if I have multiple disagreements with him. If mistakes are very rare, can he really be innocently mistaken on all those issues?