Etymology
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snug (adj.)
1590s, "compact, trim" (of a ship), especially "protected from the weather," perhaps from a Scandinavian source such as Old Norse snoggr "short-haired," Old Swedish snygg, Old Danish snøg "neat, tidy," perhaps from PIE *kes- (1) "to scratch" (see xyster). Sense of "in a state of ease or comfort" first recorded 1620s. Meaning "fit closely" is first found 1838. Expression snug as a bug in a rug attested by 1769; earlier snug as a bee in a box (1706).
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snuggle (v.)
1680s, frequentative form of snug. Related: Snuggled; snuggling. As a noun from 1901.
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cozy (adj.)

"snug, comfortable, warm," 1709, colsie, originally Scottish dialect, perhaps of Scandinavian origin (compare Norwegian kose seg "be cozy"). In Britain, usually cosy. Related: Cozily; coziness. As a noun, "padded covering for a teapot to keep the heat in," 1863.

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nuzzle (v.)

early 15c., "to bring the nose to the ground," back-formation from noselyng "face-downward, on the nose, in a prostrate position" (c. 1400), frequentative of nose (v.). The meaning "burrow with the nose, thrust the nose into" is attested from 1520s; that of "lie snug" is from 1590s, influenced by nestle, or by nursle, frequentative of nurse (v.). Related: Nuzzled; nuzzling.

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cosh (n.)

"stout stick," 1869, of unknown origin. Also as a verb, "to beat with a cosh." Related: Coshed; coshing.

Other English words of the same form, all apparently unrelated, include a provincial word for "a cottage, a hovel" (late 15c.), a provincial word for "the husk of corn" (1787), and an 18c. Scottish adjective meaning "neat, snug, quiet, comfortable."

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cuddle (v.)

"hug, embrace so as to keep warm; lie close or snug," 1520s (implied in cudlyng), of uncertain origin. OED calls it "A dialectal or nursery word." Perhaps a variant or frequentative form of obsolete cull, coll "to embrace" (see collar (n.)); or perhaps from Middle English *couthelen, from couth "known," hence "comfortable with." It has a spotty early history and seems to have been a nursery word at first. Related: Cuddled; cuddling. As a noun, "a hug, an embrace," by 1825.

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hole-in-the-wall (n.)
"small and unpretentious place," 1816, perhaps recalling the hole in the wall that was a public house name in England from at least 1690s. "Generally it is believed to refer to some snug corner, perhaps near the town walls," but the common story was that it referred to "the hole made in the wall of the debtors' or other prisons, through which the poor prisoners received the money, broken meat, or other donations of the charitably inclined" [Jacob Larwood and John Camden Hotten, "The History of Signboards: From the Earliest Times to the Present Day," 1867]. Mid-19c. it was the name of the private liquor bar attached to the U.S. Congress.
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tuck (v.)
late 14c., "to pull or gather up," earlier "to pluck, stretch" (implied in tucker "one who finishes clothes by stretching them on tenters, late 13c. as a surname), probably from Middle Low German or Middle Dutch tucken "pull up, draw up, tug" (cognate with Old English tucian "mistreat, torment," and related to Old English togian "to pull," German zucken; see tow (v.)). Sense of "thrust into a snug place" is first recorded 1580s. Slang meaning "to consume, swallow, put into one's stomach" is recorded from 1784. Related: Tucked; tucking.
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nest (n.)

"structure built by a bird or domestic fowl for the insulation and rearing of its young," Old English nest "bird's nest; snug retreat," also "young bird, brood," from Proto-Germanic *nistaz (source also of Middle Low German, Middle Dutch nest, German Nest; not found in Scandinavian or Gothic), from PIE *nizdo- (source also of Sanskrit nidah "resting place, nest," Latin nidus "nest," Old Church Slavonic gnezdo, Old Irish net, Welsh nyth, Breton nez "nest"), probably from *ni "down" + from PIE root *sed- (1) "to sit."

From c. 1200 of an animal or insect. Used since Middle English in reference to various accumulations of things, especially of diminishing sizes, each fitting within the next (such as a nest of drawers, early 18c.). Nest egg "retirement savings" is from 1700; it was originally "a real or artificial egg left in a nest to induce the hen to go on laying there" (nest ei, early 14c.), hence "something laid up as the beginning of a continued growth."

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chestnut (n.)

type of tall tree native to western Asia, southern Europe, and eastern U.S., also the large "nut" that it produces, 1560s, from chesten nut (1510s), with superfluous nut (n.) + Middle English chasteine, from Old French chastain (12c., Modern French châtaigne), from Latin castanea "chestnut, chestnut tree," from Greek kastaneia, which the Greeks explained as either "nut from Castanea" in Pontus, or "nut from Castana" in Thessaly, but probably both places are named for the trees, not the other way around, and the word is borrowed from a language of Asia Minor (compare Armenian kask "chestnut," kaskeni "chestnut tree"). In reference to the dark reddish-brown color, 1650s. Applied to the horse-chestnut by 1832.

Slang sense of "venerable joke or story" is from 1885, explained by U.S. actor Joseph Jefferson ("Lippincott's Monthly Magazine," January 1888) as probably abstracted from the 1816 melodrama "The Broken Sword" by William Dimond where an oft-repeated story involving a chestnut tree figures in an exchange between the characters "Captain Zavior" and "Pablo":

Zav. Let me see—aye! it is exactly six years since, that peace being restored to Spain, and my ship paid off, my kind brother offer'd me a snug hammock in the dwelling of my forefathers;—I mounted a mule at Barcelona, and trotted away for my native mountains. At the dawn of the fourth day's journey, I entered the wood of Collares, when, suddenly from the thick boughs of a cork-tree—
Pab. [Jumping up.] A chesnut, Captain, a chesnut!
Zav. Bah! you booby, I say, a cork.
Pab. And I swear, a chesnut—Captain! this is the twenty-seventh time I have heard you relate this story, and you invariably said, a chesnut, till now.
Zav. Did I? Well, a chesnut be it then. But, take your seat again.

Jefferson traced the connection through William Warren (1812-1888), "the veteran comedian of Boston" (and Jefferson's cousin) who often played Pablo in the melodrama.

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