Everyone has watched enough crime dramas to know that one of the easiest ways to bring down a Big Bad gangster is to catch one of his mooks doing something illegal and threaten the mook with a ton of jail time for it. After spending some time thinking about what it would be like to spend 20 years or so in jail, the mook breaks down and tells the cops what they want to know. The cops arrest the mook's boss, the mook testifies, and everything ends happily ever after.
Sometimes though, things don't go so smoothly. Sometimes the boss is so fearsome that the prospect of the boss looking to get revenge on the mook (or their family) means the mook will gladly take anything the law can throw at them rather than that. Bonus points if the Big Bad is either a Torture Technician or keeps one around just for this purpose.
Naturally, this can also apply to the general public, although in this case it may be a more general fear and unwillingness to get involved that is the motivation. Tends to be particularly common in poor countries where the criminals may have more control than the government. Things aren't helped either if the police are known for being corrupt, brutal, incompetent, or all three.
This is often the result of the Big Bad using Outscare the Enemy and I Control My Minions Through... (Fear). It's particularly likely to happen if the villain is The Dreaded or the crime organization in question happens to be Ruthless Foreign Gangsters.
A variation is for police to make it look like the perp is cooperating with them so that they'll be targeted anyway, and their only hope is to take their chances with the police.
Anime & Manga
During the Stardust Crusaders arc of Jojos Bizarre Adventure, this it the main reason why the heroes have so little information on the Big Bad's stand. The most obvious example is Daniel J. D'Arby, who has a mental breakdown after being cornered to spill this information.
In the Lucky Luke book "Billy The Kid", everyone is too scared of him to testify when he's arrested.
Common in Diabolik, with mooks and even ordinary citizens being often too scared by mob bosses to testify against them. To Ginko, the only thing most frustrating than this when dealing with them is their tendency to piss off Diabolik, resulting in the worst criminal of Clerville either killing them or giving Ginko the evidence he needs.
Mass Effect Human Revolution: Whoever Shepard's killer is working for, Harkin thinks their wrath would be worse than anything C-SEC could do. Considering double crossing the Alliance Black Ops could result in a bioroid ripping his head off, himself being replaced by one, or both, it's very justified.
Sal Maroni spells this out for Batman in The Dark Knight. Everyone, from the ordinary citizens to hardened criminals are more afraid of The Joker than of Batman. No one is going to give Batman information on the Joker, knowing what sort of reprisals would result. What makes this even more notable is that The Joker isn't even a mob boss. In fact, the mob thinks that he's working for them but they are STILL too terrified of him to give him up.
This also pops up in The Dark Knight Rises. Selina Kyle is arrested by Blake, and he wants to know what she knows about Bane, who Selina just saw break Bruce's spine. He offers her protection in exchange for information, but Selina just gives him a look of "Really?" that tells him it'd do no good. When she is placed in Blackgate prison, Bane liberates all of the prisoners.
Taken to the extreme in Steven Seagal vehicle Marked for Death, where one mook is so sure that his boss can't be taken down and so scared of him that he jumps out a window when cornered by Seagal rather than act as The Stool Pigeon. It's brought up even earlier in the movie, when the Big Bad asks his mooks "Who do you fear more, him or me?" Their wordless reaction makes the answer perfectly clear.
A Bronx Tale has the Innocent Bystander version. When the protagonist Calogero witnesses the neighborhood Mafia boss kill a man right in front of him at age 8, he doesn't tell the police anything both because of the street ethic of his neighborhood and out of fear. Later at confession he refuses to even tell the priest any details. When the priest encourages him not to be afraid because no one is more powerful than God, Calogero responds by saying "Your guy may be bigger than my guy up there, but my guy is bigger than your guy down here." The priest reluctantly concedes the point and gives up asking.
The first Once Upon a Time in China movie also used the Innocent Bystander version. When Wong Fei-Hung beats the crap out of a powerful gangster and some of his men, all the people around applaud and cheer for him. Fei-Hung then says that he's going to turn the gangster in and asks for someone to testify in court. Everyone in the crowd that had gathered around promptly leaves.
Notably averted a few times in Payback. Several people tell main character Porter (an Anti-Hero who verges on being a Villain Protagonist) that the Outfit will kill them if they talk to Porter. Porter's response is always the same; "What do you think I'm going to do to you? Worry about me."
Early in The World Is Not Enough, a character fears Renard so much that she commits suicide rather than be arrested.
Bond: We can protect you!
Her: Not from him!
In Timecop, one of the Big Bad's accomplices refuses to talk to the Time Police, because the worse they'll do to him is execute him as he is now, while the Big Bad can and will have him and his whole family Ret Gone if he talks.
This seems to be the case in The Usual Suspects when Kujan brings up Keyser Soze and Verbal reacts with stark terror. Subverted at the end when it's revealed that Verbal is Keyser Soze.
This is a recurring problem for the cops on The Wire and with good reason, as many characters who decide to become witnesses end up dead. This is particularly evident in season 1, where in order to convict Barksdale enforcer Bird they have to have Omar, (a Sympathetic Criminal with a grudge against the Barksdale group) give an obviously false testimony because all the actual eye witnesses are unwilling to testify and risk being killed for it. (Bonus points: Bird is on trial for killing a witness who testified against another Barksdale member.)
The Mentalist: one Victim of the Week is the son of a mob boss who is dying of cancer. The boss is on record as planning to kill the person responsible. Cho goes to investigate a possible suspect, who got beaten by the son's goons over a bar dispute. The suspect loudly says that the victim was a great man and he had no problems with him. Then he tells Cho quietly to come back after the old man dies & he might have a different story to tell.
One episode of Psych was only resolved in a crowd scene where the bystanders were shamed/encouraged to come forward and start testifying about the behaviors of the gang in their midst. It was a Christmas Special.
An epsiode of CSI: Crime Scene Investigation had a hitman refuse a deal to name the man who hired him in exchange for not facing the death penalty, figuring he'd live longer on death row.
On Justified a corrupt FBI agent working for mob boss Nicky Augustine chooses to kill himself rather than be arrested because he believes that merely being arrested will cause Nicky to have the agent's family killed. Nicky later threatens Raylan's family in order to get Raylan to give up a federal witness. Raylan believes him and knows that if he arrests Nicky, Nicky will make good on the threat. Instead, Raylan gets a rival mob boss to kill Nicky.
In season 3 of Dexter, the police catch on to the identity of a killer when a witness who had previously begged not to be deported away from his family does a complete turnaround at the mention of the man's name. They eventually secure his cooperation by tricking him into thinking that the killer is targeting him anyway.
An episode of Castle has the former prison mate of a suspected serial killer wary of cooperating for fear of reprisals that the man can secure through his prison connections (which seem justified when he gets beaten in the yard). It turns out that the prisoner is the serial killer, and manipulated the suspect and the police in order to get himself out of prison.
In the first episode of Lexx, an admiral in service to His Divine Shadow orders his ship to follow the Lexx into a Fractal Core as per His Shadow's orders. When his subordinate warns him that doing so would likely kill them all, he replies that since he's never gone through a Fractal Core before, he does not know that with certainty. On the other hand, the admiral does know with certainty the fate of those who disobey His Shadow's direct orders. So into the Fractal Core they go.
There's an amusing scene in Max Payne 3 when the hero, still clueless as to how law enforcement works down south, suggests they just arrest the corrupt politician dealing dope and selling organs. Gee, why didn't the DEA think of that?
An overheard audio diary in BioShock has a smuggler caught and tortured by the Rapture police say, "...Whatever Ryan thinks he can do to me, Fontaine can do double!". Considering you find said diary on his bound, electrified corpse, that's pretty damn impressive.