The law in the 1960s?

I've just seen a film on tv in which a man is accused of murder, taken to the police station, and questioned by the police without a lawyer being present. Now presumably this couldn't happen today (could it?) and I believe I'm right in saying it couldn't have happened in the 60s..? Or am I wrong? Have things changed since then? Could a man have been questioned about murder in a police station without having his own lawyer present in America in the 1960s? Many thanks.