It's not a constant occurrence, but it's been seen on a few occasions (mainly in Sit Coms): Whenever there's a foreigner who's a janitor/plumber/other mediocre job, they occasionally mention that "In old country, I was neurogeneticist/aeronautical engineer/other fancy job."
Sadly, this is Truth in Television; professionals immigrating often find it difficult to use their native accreditations to keep doing their jobs. Not to mention that law degrees usually have the unfortunate consequence that, apart from being country-specific, they're often state-specific. They don't even have to want to do the job. It is worth noting that standards vary from country to country and, for example, an engineering degree from country X is not up to the standards of country Y because engineers in X gained as much knowledge from their courses as a middle schooler from Y.
It's often believed, especially by Americans who have never been outside their country, that professionals in poorer countries immigrate to the West because they live in abject squalor in their home countries. The reason for this "interesting" belief is that a professional in a poor country may earn less than a cab driver in the United States. However, this fails to take into consideration that the cost of living is massively higher in the United States. In many poor areas it's possible for a large family to live in comfort, even luxury, on a monthly salary that's less than the average daily salary in New York. (Some capital cities in poorer countries are very expensive for expatriates, but expats have specific needs that locals rarely have - security above all.) Most professional immigrants either have a Western education and are able to enter the job force, are unaware that their credentials aren't transferable, have been sold a story by an "immigration consultant", or are escaping something other than poverty such as discrimination, political oppression, political instability, rampant crime, or war, or are seeking something other than employment (e.g. better education for their children).
Often overlaps with research failure, especially when universities from Eastern and Central Europe are mentioned. Most graduates from such countries have little trouble nostrificating their diplomas, especially if they come from renowned universities and their profession is in high demand. If anything, this trope is the result of bureaucracy or a case described below.
When it's language skills rather than job skills, the character is Eloquent In My Native Tongue. This can be a justification for Almighty Janitor.
In the second Once Upon a Time in China movie, the British Ambassador declines Wong Fei-Hung's offer of medical assistance for the people who had been injured by the White Lotus cult because he doesn't trust Chinese medicine, instead calling on the Western trained Sun Yat-sen. However, he is willing to go back on that and offer his honest thanks after seeing the results when Sun requests Wong's assistance due to a shortage of Western medicines.
The father in The Kite Runner was a noble, well-to-do politician in Afghanistan, but emigrated to America to become a gas station attendant.
And the father-in-law had been a military leader in pre-Taliban Afghanistan, but he was too proud to take an entry level job in the US, so he was feeding his family with welfare and flea-market earnings instead, waiting for the monarchy to return and free Afghanistan, where he would return to the ministry. Though not at all in the way he planned, the Afghan government is toppled, so he does return to Kabul to take his ministry post.
In Fail-Safe, a character's father had been a surgeon in Germany, but moved (fleeing the Nazis) to the United States. Because he had a family, he couldn't afford to redo his residency and ended up becoming a butcher.
In the Woman Warrior (a memoir), Maxine Hong Kiston's mother was a damn good doctor in China; when she moved to the US, she opened up a laundromat with her husband.
In one of the later Adrian Mole books by Sue Townsend he works as a chef in a posh London restaurant. He points out that all the people he knows who are in menial jobs were all doctors, professors and lawyers in their various native countries (possibly serving to underscore Adrian's relative failure, as he has no excuse for being almost as poorly employed).
In The Etched City, Raule's medical qualifications are not recognized by the medical establishment in Ashamoil, and she is forced to take a low paid job in a charity hospital.
In the Mercedes Lackey novel The Serpent's Shadow, Maya, who had been a practicing doctor for years in India, had to take her examination records to a local hospital and be interviewed by its chief surgeon before she would be allowed to practice medicine in London.
I wonder if the above description misses something. I would hope any new physician would be asked to submit board certification and pass an interview before being hired.
In the Dolphin Diaries books, the ship's cook and chief engineer was apparently a top notch engineer in China, but didn't speak English very well and could only find menial work as a cook in the US.
An inversion in Last Watch. Anton goes to Samarkand, Uzbekistan, and gets into a taxi driven by an elderly Uzbek who speaks Russian without an accent. Typical to an elderly Eastern man, the taxi driver has many deep thoughts that Anton initially assumes to be typical Eastern wisdom. However, the driver reveals that he has a Ph.D. in Psychology which he earned in Moscow. Apparently, there's no demand for a shrink in Uzbekistan, so he drives a cab instead.
Live Action TV
Heroes in the graphic novels, when Mohinder turns up searching for his dad, his quest leads him to a taxi firm.
"Dad was working here? But he was a geneticist!"
"Hey, back in Russia, I was one of the top-ranked ballerinas. In America, I'm a taxi dispatch. Want a job?"
Boy Meets World had a hopeful variation. Eric took a job as a security guard instead of trying to do his SATs again to get into college. His partner turned out to have been an engineer in wherever he came from, Eric asked him why he was reading the SAT prep book. The response: "So that I can let this country know I am an engineer."
In another episode when Eric teaches a citizenship class he says he is the smartest one in the room, and a student says that in his country, he was a nuclear physicist, but Eric tells him that, in America, everyone starts with a clean slate.
Oliver Beene once showed an Italian boy in the school's throw-away remedial class who had taken college-level courses in Italy. Being a high school example, this is probably because he didn't speak English well enough to understand what he was being taught, rather than because his previous education wasn't good for anything in the new country.
In an episode of The Big Bang Theory, Sheldon Cooper and the rest of the gang are in a physics competition. The final question baffled both groups. The correct answer was given by the third-floor janitor, who, as it turns out, was a physicist in St. Petersburg before the end of the communist regime.
In one episode of Law & Order, a doctor is on trial for Manslaughter, his Indian colleague is testifying to his actions, and the doctor's lawyer draws attention to his training in a foreign medical school. Fortunately, another Harvard-trained doctor could verify this testimony.
An episode of Scrubs featured an "intern" who had been a brilliant surgeon in his native (unnamed vaguely Eastern European) country, but upon coming to the US, was being forced to start over at the bottom of the system. His name was Miloš' Radovicnouizicioiuizcinicizich'l. Turk was jealous of his having getting to reattach a child's arm inside a rusty car.
Nahum, the janitor in Slings and Arrows, never gets his backstory fully explained, but it's suggested that he was either an actor or a director in Nigeria.
In Big Love, season four, Ana was a dental hygienist back in her home country but was having difficulty passing the test in America because her English was not so good. Ergo, she was working as a waitress in a diner where she met Bill (and Margene).
In Vanishing Son the two boys try to get jobs in the US after defecting from China. Despite one having 3 1/2 years of college majoring in Biology and the other a degree in History, the only jobs they can get are gutting fish.
An episode of Doctors had an Asian bus driver who saved the life of a boy on his bus, then feigned a lack of English to avoid getting in trouble for practicing without a license. So they called in a translator, which led to a Crowning Moment of Funny when she recognised him:
Doctor: You know this man? Translator: Yes, he's my husband. And an idiot.
Zig-zagged by Jeff from Community, who has to go back to college because his undergraduate degree was from a university in Colombia, not Columbia as he had stated. While he did earn his law degree legitimately (passed all relevant coursework), the fact that he had misrepresented his undergraduate credentials raises the question of whether he got into the law school fraudulently.
In That '70s Show, Jackie mentions her maid could have been a doctor in her country.
In The Sopranos, the husband of the Sopranos' maid was an engineer in Poland.
In Desperate Housewives, a house cleaner teaches Juanita while she's being home schooled. She had a doctorate in engineering from the University of Bucharest.
In True Blood, Yvetta, one of the strippers in Fangtasia, reveals that she was a Cardiologist back in Estonia.
A one-time game in Whose Line Is It Anyway? parodies the "This Is Your Life" type of reality gameshow, and Colin Mochrie, as a voice from Wayne Brady's past, rants about how Wayne took his job as a sewer worker. Wayne follows up by claiming that Colin is a Mexican immigrant (despite not having the accent), whereupon Colin claims to have taken "four years of English literature".
Hide from The Office was a renowned heart surgeon in Japan, but is a warehouse worker in the States. (However, he's probably laying low because he screwed over the Yakuza.)
Frasier. After Dr. Schachter gives the Crane brothers a virulent "The Reason You Suck" Speech regarding their Sibling Rivalry ("That is it! That is it! In thirty years as a couples therapist, I've never said what I'm about to say: Give up! It's hopeless! You are pathologically mistrustful of each another, competitive to the point of madness! So, trust me, just meet each other at weddings and funerals, and the rest of the time, stay the hell away from each other!"), the boys agree on his prognosis, but then use his degrees as an Ad Hominem argument to ignore his advice, even though they admit Martin says the same exact thing, and kick him out of the building!
Frasier: Well, there's no arguing with Dr. Schachter's credentials. My God, the man is an expert in his field. He graduated from the University of... Grenada!
Niles: [shocked] Well, surely that was just his undergraduate schooling.
Frasier: Oh yes, of course, his graduate work was done in... Aruba!
Niles: An all-Caribbean schooling... well, tally me banana!
Subverted/parodied in PhoneShop, regarding the shop's cleaner;
Lance: Janine! Back in Nigeria that man's a leading neurosurgeon! Janine: Really?! Lance: Nah, he's a fucking cleaner.
Featured on ER where recurring foreign nurse "Bob" (not her real name) has to save a patient when all the doctors are busy. Turns out she's a cardiac surgeon from Eastern Europe. The other nurses are horrified as they don't know that she knows what she's doing, and she's horrified she's blown her chance at the hospital, or even to stay in the country. Dr. Carter assures her that's not so.
Dr. Elizabeth Corday has to restart her residency, and she's from the UK!
Used in Castle, where it's revealed that a murdered cab-driver was, in his home country, a high-ranking nuclear scientist. It's eventually revealed that he could have put his talents to service in America when he defected, but was disillusioned by working on a nuclear weapons program in his home country and was happy enough working as a cabbie and living with his family.
This may be also because people applying for a position with a government contractor have to pass through a grueling background check. Immigrants are almost always rejected on suspicion of divided loyalties.
A character in the Endgame episode "Turkish Hold'em" was a doctor back in Turkey, but a garbageman in Vancouver.
Played with in Arrested Development. Lindsey tries to break up a strike at the construction site by driving up with a busload of guests from Lupe's family reunion and announcing, "You won't work? Fine! Here are a bunch of Mexicans who would love to have your jobs." One of the guests protests that he's a professor at the University of Mexico City.
A one-time character in Person of Interest was a first-class surgeon working as a coroner because he couldn't afford to get licensed to practice medicine in America after immigrating due to the fact he sends most of his paycheck back to help his family. Finch resolves this by giving him a Briefcase Full of Money in exchange for him treating a critically injured Reese without reporting the gunshot wounds to the authorities.
Elementary had a murderer who the police had determined had to be both a) a doctor or other medical professional and b) fluent in Ukrainian, which none of the doctors at the hospital they were investigating spoke. It turned out the janitor was a Ukrainian immigrant who had been a doctor in his home country.
Heavy Weapons Guyhas a degree in Russian Literature. In a deviation, he's actually very happy with his current job, which allows him to shoot people with a very big gun; apparently, Russian Literature is just what he studied because what the hell. His poor English is implied to just be a lack of fluency; in the Russian version of the game, his speech takes a much more sophisticated, educated tone.
And when asked whether that comes inhandy in his line of work, he replies "More than you think."
Also, Russian literature tends to be useless when the only people who appreciate it are under a government that wants you dead.
Family Guy one of the Portuguese fishermen (who, oddly enough, have thick Brazilian accents) employed on Peter's boat claimed that back in Portugal he was a cardiologist.
In The Simpsons, it is revealed that Indian convenience store clerk Apu graduated as a top-of-his-class computing scientist back in Calcutta. Then again, it's also implied that he could get a better job if he wanted; he's just really devoted to working at the Kwikimart.
Groundskeeper Willie was doctor in his native Scotland. When he came to America, they took away everything, including his last name.
Also Apu have US Ph.D. so his degree was at least accepted by university...