Partially a case of Truth in Television, the condition is known as Avoidant personality disorder. These people have a crippling fear of rejection (though not in the same way as the similarly-named trope) and constant worries of looking foolish or bad, and closely monitor everything they say and every reaction from anyone they're speaking to (on the rare occasion that they even do speak to others). This often results in broken speech patterns and a lack of fluency, due to having their attention split between tasks. These people will often wind up choosing to be lonely, rather than risk rejection. However, where the trope fails the reality test is in the "growing out of it" part. In reality, the condition usually just becomes worse and worse. Sufferers rarely seek help, either because they feel that they don't want to waste a therapist's time, or they are worried they'll be rejected by them (although, in TV land, there's another reason). In a painful irony, many sufferers, due to their lack of speaking or participating in activities, are viewed as aloof or arrogant, when really they're so nervous when they're spoken to that they may be rendered incapable of speech. This is the most severe form of this personality disorder; there are plenty of others who remain functional, but very shy, isolated from society and unhappy with their situation. With a proper support network involving family and friends, such people can be coaxed into therapy, where counseling and medication can alleviate many of the symptoms.
A more common and less severe version that also results in people that fit this trope is Social Anxiety Disorder, also known as Social Phobia (there is in fact controversy over whether or not social anxiety disorder and avoidant personality disorder should be considered separate disorders or merely varying severities of the same disorder). Encountering unfamiliar people or being placed in a situation where the sufferer is possibly subject to scrutiny leads to intense anxiety or panic attacks.
"A painfully shy boy, Coolidge would go into a panic at the sound of a stranger's voice in the house. Writing in a letter to a friend years later, he recalled that when visitors would sit with his parents in the kitchen, he found it difficult to go in and greet them. "I was almost ten before I realized I couldn't go on that way. And by fighting hard I used to manage to get through that door. I'm all right with old friends, but every time I meet a stranger, I've got to get through the old kitchen door, back home, and it's not easy."
Lord Henry Cavendish, an eighteenth century English scientist and recluse, was so painfully shy that he communicated with his housekeeper by letter. He would, rarely, attend parties. Guests who wanted his honest opinion were advised to approach and speak, "as it were, into a vacancy" and hope to hear a mumbled reply. More likely they'd hear a mortified squeak and turn to find he'd fled. However, his seclusion left him the free time to be as meticulous as humanly possible and thus he was the first to discover the gravitational constant. (It required measuring the minuscule torque on a wire caused by the gravitational attraction of two metal balls. From another room. Through a telescope.) Unfortunately both for Cavendish and for science, he also didn't publish his findings, which often meant that a) others got the credit, and b) those findings were delayed, sometimes by decades.
How many findings? Well, he was born in 1731 and died in 1810. Years later, James Clerk Maxwell found his writings and affirmed he was the first to independently discover: Richter's law of reciprocal proportions (Richter published it in 1792), Ohm's Law (1827), Dalton's law of partial pressures (1801), Coulomb's Law (1785), Charles's Law (1802), and written a treatise, "Heat", which, after being studied this century, has become the most detailed and correct writing regarding thermodynamics from his age. Among other things.
According to historians, young Louis XVI of France. He was described as shy, having no social graces, and very hesitant.
Actor Colin Morgan is very shy. He comes across as reserved, blushing and very hesitant in most of his interviews. This adds to his Moe and Adorkable appeal, as even his fellow star admitted to being jealous by all the attention he receives for this.
Bradley James: "Everyone loves Colin "Awww he's so nice, so innocent. You know aww Colin aww. But Bradley, don't worry about Bradley, he's fine, he'll, you know... whatever. Colin awww, we'll look after Colin, awwwww."
Another scientist example is that of the Chinese mathematician Chen Jingrun. His first job after graduating from the math major—teaching high school math— only lasted for a single year because by the time he already had two Heroic BSODs that required hospitalization. He was transferred (remember, we're talking a period when China was a Soviet clone) to the post of an university librarian, which, ironically, jump started his mathematician career...
Comedian Mitch Hedberg's signature style (wearing sunglasses, hunching over the microphone and spending most of his acts staring at the floor) was due to a crippling stage fright. Those mannerisms helped him forget that an audience was watching him.
Mahatma Gandhi actually started out like this. In his first case as a lawyer, he entered the court, was too afraid to say anything, and then fled in terror. (He later refunded his client's fees.) Over time, he grew out of it, although he remained rather reserved.
Swedish singer Sally Shapiro (real name unknown) has never toured except for once in 2008, rarely gives interviews, and has a very Moe appearance and voice.
Harrison Ford originally took up acting as a way to help with his social anxiety and fear of public speaking. Even now in interviews, while friendly, he is still fairly quiet.
Toshi of X Japan is anything but quiet on stage, but when it comes to interviews he mostly leaves the talking to band leader Yoshiki. There is one interview where he's speaking very clearly but...well, take a look for yourself. Even more so when they're doing interviews in English. (Toshi can actually speak better English than Yoshiki, but really seems to have trouble conducting interviews while speaking it.)
Actually, every member of X Japan besides Yoshiki could be considered this.
During R.E.M.'s early years, frontman Michael Stipe was painfully shy. So much so that when the band appeared on Late Night with David Letterman, Stipe actually hid behind Peter Buck so Letterman wouldn't talk to him. He's outgrown it almost completely, though.
Henrik Ibsen was this all the way. He was a loner at school, and later in life he had problems with closeness. It is said that pr 1899, he and his colleague Bjørnson was invited by the king to a diner. It was observed that the two authors were in different rooms. Bjørnson, in one room, sparkled in a crowd of listeners, while Ibsen sat in another room, huddled over his food in a corner, with his back to the audience. A Shrinking Violet at the age of 71 years.