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The answer to this question has two components.
To begin with, we can assume that the “Brains in a barrel” argument itself is correct. In this case, Putnam did not refute the very possibility of a conditional Matrix, but blocked us from meaningfully talking about it. The argument goes like this: even if we are actually a brain in a barrel, we cannot meaningfully say that we are a brain in a barrel, because we will not be able to make a reference beyond the illusory world “inside the barrel” to the real world and us as a brain in this real world (the latter fact is connected with the rigid defense of the causal theory of reference). So it's quite possible that we're actually hanging out in a vat, but it's impossible to say that correctly. And if this cannot be meaningfully stated, then we can only abandon the hypothesis itself.
This is if the argument itself is correct. But it is quite possible that it is wrong. In particular, it can be noted that the aforementioned Kripke-Putnam causal theory of reference plays a key role in this argument. First, if it is incorrect (on some independent basis – for example, there is a hidden contradiction in it), the whole argument falls apart. Secondly, its correctness can be doubted on the basis of this argument itself. The causal theory of reference-like any theory of reference – is an empirically unverifiable theory; in other words, it is unverifiable. Hence, the degree to which it is correct should be determined on the basis of at least two factors other than verification: consistency and the ability to convincingly describe all the phenomena for which it was intended to describe (i.e., the phenomena of reference). However, Putnam behaves as if the causal theory of reference is already an independently confirmed theory, i.e., as if it were one of the empirically verified theories of natural science, from which, after verification, it is possible to extract consequences that can also be considered true. The brain-in-a-barrel argument is just such a corollary from the causal theory of refraction. But this is a counterintuitive consequence, as Putnam himself says. Counterintuitive consequences from the theory would be acceptable if the theory itself was independently confirmed, so that we would have clear reasons to consider it true regardless of these consequences (many purely theoretical consequences of modern physical theories are quite counterintuitive), or if we had a way to empirically verify the truth of these counterintuitive consequences themselves (as in the case, for example, with the Poisson spot). However, in Putnam's case , neither the theory nor its implications can be independently verified for truth. In this case, I think we can assume that the very fact that the theory of causal reference provides a counterintuitive consequence should be considered as evidence against this theory itself: the theory of causal reference does not cope with describing a certain situation of reference – and its value, like any other theory of reference, is solely to be successful in describing the entire possible set of cases of reference.
So, in general, the answer is: no. If the argument is correct, then no, Putnam did not refute the possibility of the Matrix, but only forbade reference to it. If the argument is incorrect, then even the last one failed.