You are asking whether these theories are [$\omega$-inconsistent](http://en.wikipedia.org/wiki/%CE%A9-consistent_theory). Of course, we all think that our theories are $\omega$-consistent, because we want and expect that the natural numbers referred to inside the theory are somehow the same as those in the meta theory. But of course, because of the incompleteness theorem, we cannot prove this within the theory itself. A theory is $\omega$-consistent just in case it has an $\omega$-model, a model in which the natural numbers of the model are the same as the natural numbers of the meta-theory.

Nevertheless, our stronger theories can often prove that our weaker theories are $\omega$-consistent. For example, ZFC proves that PA is $\omega$-consistent, and ZFC+large cardinals proves that ZFC is $\omega$-consistent. My own view is the assertion that a theory is $\omega$-consistent is just one stop along a continuum, from assuming consistency up to iterated consistency up to having a well-founded model up to large cardinals and so on, moving up in consistency strength at each step. 

Meanwhile, if these theories are consistent, then it is easy to make examples of $\omega$-inconsistent theories extending them. For example, the theories PA+$\neg$Con(PA) and  ZFC+$\neg$Con(ZFC) are consistent by the incompleteness theorem, but not $\omega$-consistent, because each of them asserts that there is a certain proof of a contradiction, which cannot be instantiated by any actual natural number witness.