This is quite possible, that a theory $T$ is inconsistent but any deduction takes so long that we do not know. 

Hugh Woodin has a short, nice paper, that I recommend you take a look at, where he addresses the possibility that (a fragment of) Peano Arithmetic (PA) is inconsistent, but any inconsistency is too long for us to be able to detect it. The paper is "The Tower of Hanoi", in *Truth in mathematics (Mussomeli, 1995)*, 329–351, Oxford Univ. Press, New York, 1998.

Part of his point is that although people discuss the possibility of inconsistency of large cardinals or strong set theoretic axioms, 

> "there are limitations on the extent to which our experience in mathematics to date refutes the existence'' of certain sequences of natural numbers with 'undesirable' consequences. (pg. 330)

He uses the idea of the tower of Hanoi to construct a sequence showing that exponentiation would be ill-defined, starting from such an assumption on the inconsistency of PA. (Actually, he uses that PA is bi-interpretable with what we call ZFC${}^{fin}$, so we can argue about sequences very much in combinatorial terms without worrying much about coding issues.)

Now, strong theories such as PA of ZFC can prove the consistency of all their finite fragments. Of course, the proof of the consistency of a fragment tends to use axioms that are outside of that fragment, so we are not violating the incompleteness theorem in this process. However, the experience we have gained from the analysis of this local property hints at what Gowers suggests in his comment, that we can still obtain a meaningful local theory even if the global version makes no sense. 

Since it would turn into a bit of a quagmire to carry this discussion with PA, for simplicity here I am simply assuming PA is consistent, but let me clarify this "meaningfulness" somewhat in the context of ZFC. *Most* of what we do with ZFC can actually be carried out in the theory where replacement is restricted to $\Sigma_2$-formulas, and there is reasonable 'evidence' that if ZFC is inconsistent, then its inconsistency comes from an instance of replacement applied to formulas of larger complexity than $\Sigma_2$. This means that a non-negligible fragment of our intuition about models of set theory would actually be correct, only that it would not be about ZFC, but about this restricted form.

Inconsistencies would simply not affect that part of our understanding, and if they ever were to surface, we could probably do damage control in a less panicked fashion than usually feared. We would in fact, in hindsight, see that the inconsistencies would explain how our intuitions about the $\Sigma_2$-case simply cannot carry to larger fragments, and for day-to-day practice, what most of us do would be completely unaffected.

[That said, of course I should add the usual disclaimer that I do not presently believe ZFC is inconsistent, so whatever I say may be considered suspect.]