Marty wrote:

I don't understand why you can fail to believe 2+2=4. You could think 2+2=4, and incidentally have another belief that contradicts that (2+2=5). It'd be certainly odd, but not impossible.

So what? That isn't an example of your belief that 2+2=4 failing. That's an example of your beliefs being inconsistent.

The point is, this proposition is true: "For some person who believes 2+2=4, it is not possible that that belief fails." But that proposition is also a counterexample to the principle that nothing can be a belief unless it can possibly fail.

Marty wrote:

And the proposition 2+2=4 is surely necessarily true, but I'm not sure we can consider such a proposition outside of our own beliefs to be a belief.

I'm not considering the proposition 2+2=4 to be a belief. I'm pointing out that if it is necessarily true, then any person's belief that 2+2=4 cannot be wrong. The proposition and the corresponding belief are not the same thing, but the latter cannot fail when the former is necessarily true.

Marty wrote:

Even a-priori analytic statements can fail. I can even fail to believe that I believe.

Sure, but that's not relevant to the counterexample.