Most people don't realize that what you see on t.v. is actually true, anything you say can and will be used against you. Often, police officers will come to a suspect to get his/her story of what happened to lock them in to a version of events that is almost always incriminating and fails to adequately explain a valid defense like self-defense. Its obvious there is more to the story and without an admission of guilt, the officers realize they might not have the best case. They might be building a case or waiting for you to build it for them by talking. They can come back often, however if it gets to the point of obvious harassment you can take legal action, or simply get in touch with the right lawyer to let them know you are represented and diffuse and questions to the lawyer. Its a good idea to contact a lawyer to take the offensive and try to prevent a false charge from being brought and begin building the best defense if you or "the man" are charged.
It is advantageous to the police to talk to a witness or suspect before issuing an arrest warrant. It allows the police the option of making it seem like if you talk, they might not charge you, put in a good word with the judge, or help you in some way, all of which is almost universally untrue. If there is no arrest warrant, you may also be held for 72 hours before they have to release or charge you. In other words, they can let you sit in a jail cell to see if you will cave in and talk. The best way to deal with the situation is to hire an attorney. Your attorney will bring the situation to a head such that the police will either have to charge you with the evidence they have, or leave you alone.