Can you get into trouble for not answering questions when talking to the police? Or is this a legit cheat code that can get you out of trouble? Which one is it?
There are certain situations where the law may require you to provide some information -- particularly when you've been pulled over for a traffic violation -- but it is typically limited to basic identifying information.
Beyond that, it is legally advantageous to refuse to answer questions. If you are doing so, though, you should explicitly invoke your Fifth Amendment right against self-incrimination, and your Sixth Amendment right to counsel. Invoking the Fifth prevents the police from using your refusal to answer as evidence, and invoking the Sixth requires the police to stop asking you questions.
It may, however, be practically disadvantageous to refuse to answer questions, as there are cops who are just looking for a reason to act like a psychopath.
The standard caution when given by British police upon arrest or questioning of a subject is:
“You do not have to say anything. But, it may harm your defence if you do not mention when questioned something which you later rely on in court. Anything you do say may be given in evidence.”
The linked government page explicitly says (emphasis mine):
The police may question you about the crime you’re suspected of - this will be recorded. You don’t have to answer the questions but there could be consequences if you don’t.
They actively say that not mentioning something might harm your position.
Can you get into trouble for not answering questions when talking to the police?
NO, Maybe, Yes
It depends on what, if any, legislation causes the questions to be asked whether failure to answer can get someone in to trouble.
Apart from general day-to-day conversations, there are three instances when an officer may ask someone questions:
Stop and Account. This interaction is totally voluntary and failure to answer is not, in itself, cause for reasonable suspicion of any offence, so one cannot "get in to trouble" by staying silent.
What are you doing?
Why are you in the area?
Where are you going?
What are you carrying?
There is no legal requirement or obligation to answer any of these questions, and the police cannot lawfully detain anyone to ask them.
Under Caution. To add to the answer given by @moo, above, the caution is given before questioning a person concerning their suspected involvement in a suspected offence. They do not have to be under arrest as England and Wales allows for voluntarily attended interviews under caution. One may "get in to trouble" by not answering if the trial judge directs the jury to consider the Adverse Inferences which, to use moo's words, means "not mentioning something might harm your position".
Compulsion. There are a few occasions when a person may "get in to trouble" and commit an offence by not answering a question. For example, a Disclosure Notice under s.62(3) of the Serious Organised Crime Act 2005 may require (ie compel) someone to given an answer - the offence of not doing so is at s.67. As this piece of legislation is designed, in part, to elicit information from a bit-player on the fringes of criminality in order to build a case against a "bigger fish", s.65(1) provides a statutory immunity from self-incrimination relating to the answer unless they have lied or failed to properly answer.
Unlike the situation in the UK, you have no obligation to answer questions posed by police, apart from some state requirements to identify during a traffic stop (which can be complied with by providing you license, as in the video), or in some states in general you may be required to provide your name. A judge may order you to answer questions, the police cannot. This is basically what it means to "take the fifth" – given the risk of self-incrimination involved in answering questions posed by the police, refusing to answer questions is a fundamental constitutional right in the US. It is not legal for the prosecution to introduce your refusal to answer questions at a trial, nor it it a legal basis for an arrest (does not constitute probable cause).
Get into trouble? Sure!
Get out of trouble? Maybe ...
Various jurisdictions have different implementations of the protection against self-incrimination. This may allow a suspect not to answer anything, or just not to make any statement to the facts. So if a truthful statement could incriminate the speaker, it might be wise not to say anything beyond "I want to talk to my lawyer."
Then there are situations where the police stops a motorist (or pedestrian) and all they want to do is to give a verbal caution or admonishment. Or where the police see someone who might match the description of a suspect. A polite and coherent answer might resolve the situation without further hassle. "I don't answer questions" could either make the police give up and bother someone else, or it might make them write the ticket, or even worse. The outcome depends on police-community relations and (unfortunately) the skin color of the persons involved.
So saying nothing isn't a "legit cheat code" to get out of trouble. It is raising the stakes and bluffing, in the hope that the cop will fold. Where I live, and for someone who looks like me, being polite and reasonably cooperative is the best strategy for minor traffic stops. That won't prevent a ticket if a ticket is clearly justified, but it makes the experience less bothersome for all sides and it could perhaps make the difference between a ticket and a warning.