Yes, they can.
Most people who grow up in a peaceful, law-abiding society are raised to trust the police to look after the well-being of its citizens. Such trust is well-placed, because the police perform a vital service in our community, and they do so very effectively.
However, if you are the person the police are investigating, you have to be very careful. When police are investigating a crime, they are permitted to use whatever tactics they have at their disposal within the limits of the law. In the course of an interrogation, or “interview” as police prefer to call it, they will often try various approaches to persuade a suspect to confess, or at least to provide them with inculpatory information they can use to seek a conviction. Sometimes these approaches are psychological: appealing to the suspect’s conscience, or using the intimidating approach of the “good cop/bad cop” routine. A good investigator will also be a seasoned performer in the art of the “bluff”, in order to persuade the suspect that silence is futile. They can do so by telling their subject that they have collected evidence that points to his guilt; or that an accomplice has confessed. They can even lie about the legal process or the consequences of the crime, in order to scare the suspect into cooperating, or at least talking. It’s up to you to be smart enough to take your information from your lawyer, whose job it is to help you, not from the police officer whose job it is to convict you.
How far the police are allowed to go in telling lies to a suspect is an issue left to the courts. But the Supreme Court of Canada has repeatedly made it clear that in investigating crime, police will be given wide latitude to engage in tactics that might not be acceptable at a dinner party or a company baseball game. How far the police can go in seeking to “persuade” a suspect to talk will depend on many factors, including the crime being investigated, and the vulnerability of the person being questioned (such as a child or person with diminished mental capacity). The scope they are permitted might surprise many people.
Short of provable abuse of the law, there is only one counterweight to this police power: the right to remain silent. The police have the right to investigate, and the suspect has the right not to give them what they are looking for. The right to counsel is related to this, in the sense that by speaking to a lawyer, a suspect being investigated by police will be more informed about the process he is being subjected to, and his rights and obligations during that process. But a lawyer cannot stop the investigation; and in Canada, unless the suspect is a youth, the lawyer is not permitted to sit in or assist his client during an interrogation. Your lawyer can arm you with information and advice, but it is up to you to protect yourself by following that advice.
During an interrogation, the police hold all the cards: they do not have to share any of the information (unlike later during the court process), and they can try various tactics to persuade you to talk. And when the investigation has reached that point, it is very rare that anything you say will stop the case from proceeding, or charges from being laid.
If you are under detained by the police investigation, except in very rare circumstances, anything that you have to say that will help you avoid a conviction will best be brought out at a later time, when you know what information the police has, and when that information can be presented in the most effective way, to the person who is most likely to pay proper attention to it: the prosecutor, not the police.
The best defence in such a situation is simple: Shut Up!