Brookline 10th-grader Ashley Olafsson sleeps with her cellphone under her pillow so she doesn’t miss “emergency’’ texts — “like if a friend broke up with her boyfriend.’’ Stephanie Kimball of Waltham, 14, is also available for urgent overnight correspondence, such as, “Hey, seeing if you’re awake.’’ Dedham ninth-grader Courtney Johnson gets as many as 100 texts while in bed. “I just don’t feel like myself if I don’t have my phone near me or I’m not on it,’’ she said.

Sure, all that middle-of-the-night communication leaves them tired, but as Olafsson explained, “It’s impolite not to respond if someone is coming to you with their problems.’’

With teenagers sending and receiving an average of 3,276 texts per month in the last quarter of 2010, according to the most recent statistics from the Nielsen Co., it’s no wonder that Michael Rich, director of Children’s Hospital Boston’s Center on Media and Child Health, is starting to see young patients who come in exhausted by being “on call’’ or semi-alert all night as they wait for their phones to vibrate or ring with a text.

He and his patients’ parents were initially baffled by the children’s increased sleepiness because bedtimes hadn’t changed, he said. “Who would think to ask a kid, ‘Do you sleep with your phone under your pillow?’ To us, it sounds like torture.’’

Children who text late into the night do not fall asleep as well, he said, and they don’t enter the deep sleep of Stage 4 REM sleep, “which is crucial to moving experiences and lessons of the day from short-term into long-term memory — in other words, completing the learning process.’’

Anticipating texts, Rich explained, leads to a bad night’s sleep in the same way as an early morning flight or other predawn obligation. “You’re so focused on not screwing up your wake-up that you don’t sleep as well.’’

A Pew Research Center study from 2010 reported that more than four out of five teens with cellphones sleep with the phone on or near the bed, sometimes falling asleep with it in their hands in the middle of a conversation. Pew researchers did not ask whether the phones were on, but Amanda Lenhart, a senior research specialist, said “many expressed reluctance to ever turn their phones off.’’

Researchers at the JFK Medical Center sleep laboratory in Edison, N.J., found in a 2010 study that teens sent an average of 33.5 e-mails and texts overnight and that their sleep was affected. A National Sleep Foundation study released this month found that almost one in five teens ages 13-18 are awakened by a phone call, text message, or e-mail at least a few nights a week.