The police are trained to speak with authority, so it's easy to believe them when they tell you what your rights are. The fact of the matter, though, is that they aren't always right. Everyone knows that people make mistakes, and police officers are people, too.
For example, according to the Washington Post, just recently a Sheriff from Phoenix, Arizona admitted that his office had mismanaged over 400 sex-crimes investigations. Think of all the people's lives affected by that error, and that's just one office.
Government officials of all varieties frequently tell you what you can and can't do, but they aren't lawyers, and they don't always know the law. It's your job to know your own rights so that you can't be taken advantage of by a system that doesn't care about you, but the law is complex.
That's why it's so important to have a good attorney: not just so that they can represent you in court, but so that they can counsel you on your rights. Unlike government officials, good attorneys know the law inside and out, and you get to benefit from that knowledge.
With a good attorney, instead of proving that you were following the government's rules, you can make sure that the government is following its own rules. The sad truth is that not everyone in this country gets a fair trial, and that's why you need to make sure you have a representative who can teach you your rights, and then stand up for them!