Henning Makholm points out one way to achieve this. But it is also the case that, in reasonable cases, "$T$ is incomplete" is independent of $T$ itself. Let's just stick to PA; the explanation will show what properties of PA are actually required.
First, we have to work out what we mean by the statement "$T$ is complete" in PA. Since PA cannot talk about models, the definition has to be syntactic. I will take "$T$ is complete" to mean that for every $\phi$, either $\phi$ is provable or $\lnot \phi$ is provable. That is, $(\forall \phi)[\text{Pvbl}(\phi) \lor \text{Pvbl}(\lnot \phi)]$, where Pvbl is the formalized provability predicate for $T$. I do not assume in the definition that only one is provable, just that at least one is; so I am taking a very weak syntactic definition of completeness. But the same argument will go through if we define "$T$ is complete" to mean that for every $\phi$ exactly one of $\phi$ and $\lnot \phi$ is provable. That is the usual syntactic criterion for completeness (although it implicitly assumes consistency in a certain sense).
If PA proves "PA is not complete" then PA proves there is some $\phi$ such that neither Pvbl($\phi$) nor Pvbl($\lnot \phi$). Thus PA proves $(\exists x)\lnot\text{Pvbl}(x)$. But it is known that $(\exists x) \lnot \text{Pvbl}(x)$ is not provable in PA because that sentence is equivalent to Con(PA) over PA. So PA cannot prove "PA is not complete".
Now assume PA proves "PA is complete". Then for every actual formula $\phi$, PA proves that either Pvbl($\phi$) or Pvbl($\lnot \phi$). However, because PA is $\omega$-consistent, if PA proves that a formula is provable then the formula really is provable. Thus, if PA proves "PA is complete" then PA really is complete. But that is not the case, so PA cannot prove that PA is complete.
In the end all that we need to know about PA is that the incompleteness theorems apply to it and that it is sound for $\Sigma^0_1$ sentences. There are certainly finitely axiomatized theories that have those properties, for example NBG set theory.