I can't comment on the general opinion, but I can point you towards this lucid and, I think, fairly even-handed discussion of a collection of essays by Putnam on the fact/value distinction: Alexi Angelides' "The Last Collapse?". Angelides reads Putnam's argument as essentially pragmatic in character (when it succeeds), and so his argument would not amount to "proof," so much as a spur to further thought about the problem.
Speaking of Putnam directly, it seems to me that Putnam attacks the distinction in quite a different way than you suggest in your question. You say that your teacher said that Putnam argued that "ethical judgments are in fact connected with cognitive judgments and therefore it is possible to have solid grounds for some ethical claims." That suggests that somehow this entanglement of fact and value allows value judgments to be given a firm factual foundation. That's certainly something people have tried to do; attempts to found ethical reasoning on evolutionary biology come to mind. But I don't think that's what Putnam is up to.
I'm not an expert, but it looks to me like Putnam is actually saying quite the opposite. Take this passage from Angelides' review (which I quote here since I don't have access to Putnam's own words). He summarizes one of Putnam's centural arguments as follows:
While the natural objects of the natural sciences are determined extensionally and explained causally, the way in which a scientific theoretical apparatus supports such explanation cannot be causal. Each theory—and the selection of a particular theory over and against its competitors—presupposes normative values such as coherence, simplicity, explanatory power, and so on. Thus a scientific theory is itself supported by the selection of values implicit in its practice (31).
According to Angelides, Putnam doesn't seem to be talking about how we can found value judgments on factual judgments. He's actually suggesting (in a rather sly way!) that factual judgments are founded on value judgments. The things that allow us to prefer one set of theories over another are themselves not facts but "normative values," to use Angelides' phrase. These normative values are of the same sort as those that motivate our ethical judgements -- at least broadly speaking. According to this line of reasoning, a particular theory may be simpler, more coherent, or more powerful without necessarily being any truer. Nonetheless we prefer those theories because we value simplicity, coherency, and explanatory power.
So I think it would be fair to say that Putnam is rejecting the fact/value distinction "from the other side," so to speak. It's not that values can be reduced to facts; it's that values enable us to produce facts! That might be an oversimplification of Putnam; I haven't read a lot of his work. But it sounds right to me. (I hope anyone with greater familiarity with his thought will correct me.)