In Numbers 11, we have the story of God's miraculous provision of quail for the children of Israel, who were tired of eating manna all the time. Although the quail was provided in massive quantities, the Israelites, according to the standard translation of v. 33, did not so much as eat a single bite before the Lord punished them with a plague: "But while the meat was still between their teeth, before they chewed it, the anger of the LORD burned against the people, and the LORD struck the people with a very great plague" (NET Bible and many others).
The phrase translated "before they chewed it" is טרם יכרת, literally, "before it was cut off." It strikes me as unlikely that the Niph'al of the root krt would mean "chew"; the default gloss for the verb is "be cut, cut off." The Aktionsart of the verb is normally telic (a Vendlerian "achievement") and not iterative (an "activity"). But words develop their little quirks and it's not out of the question that something like "chew" (or "bite") could develop out of "cut off" (and compare the NEB "they had not so much as bitten it"). Nevertheless, this verse seems to be the only one where נכרת is translated "chew."
The ancient versions, however, unanimously render the phrase with words meaning finish, be over: "While the meat was still between their teeth, before it failed" (πρὶν ἢ ἐκλείπειν, LXX), "before it ran out" (nec defecerat, Vulgate), "before it stopped" (עד לא פסק, Onkelos), "before it went away" (ܘܥܕܠܐ ܥܒܪ., Peshitta). The ancient translators apparently took the phrase to refer to the month-long period that the quails were available for eating (Num. 11: 19-20), so that an overall paraphrase of v. 33a would be "While they were still (daily) eating the quail, before the supply ran out ..." (Rashi mentions both interpretations but favors the translation of Onkelos.) This makes a lot more sense to me, since the episode presupposes that the Israelites consumed a lot of quail.
There does not seem to be any straightforward reference to "chewing" in the Hebrew Bible. The animals that "chew the cud" (Lev. 11:3 and elsewhere) actually "bring up" (מעלה), regurgitate, the cud. The standard verb in later Hebrew (including Modern) meaning "to chew" is לָעַס, which, as far as I know, is first attested in the Mishnah. I have a hunch, though, that Hebrew speakers chewed things before then and probably referred to the act with the same verb.