The expression “Sleep tight, don’t let the bedbugs bite” is often said to refer to colonial times, when children slept on rope beds underneath their parents’ beds. The ropes would be tightened for support — thus “sleep tight.” Too bad the bugs haven’t gone the way of the rope bed.The next day I heard it on "Fresh Air," where Terry Gross was interviewing a "professor of urban entomology" named Michael Potter. The rhyme, he solemnly told listeners, dated to the 1500s or 1600s, referring to the rope lattice that supported mattresses back in the day.
Not very likely. The OED says "sleep tight" just means "sleep soundly," and the skeptical view of the faux etymology has been available for some time at Michael Quinion's World Wide Words and at The Phrase Finder, which note that "sleep tight" is not really very old. Their earliest citation comes from Susan Bradford Eppes, 1866: "Goodbye little Diary. 'Sleep tight and wake bright,' for I will need you when I return." And no bedbugs appeared till the 20th century, they said.
But there was newer news, it turned out, on "don't let the bedbugs bite." The day after the "Fresh Air" broadcast, New York word sleuth Barry Popik weighed in at his blog with a wealth of additional "sleep tight" citations. He found the pests (called bugs, buggers, skeeters, and mosquitoes) infesting the bedtime sentiment as early as 1881, just 15 years after the earliest "straight" version: "'Good-night, sleep tight; And don’t let the buggers bite,' said Fred."
In any case, there's no reason a bedbug discussion should divagate into the origins of "sleep tight," since the phrase has nothing to do with repelling insects. It seems pretty clear from Popik's list of Google cites that the buggy versions were earthy variations on a sweet Victorian sentiment, coined for no better (or worse) reason than shock value and a snappy rhyme.