The constitutional floor vs the state ceiling
The U.S. Constitution guarantees a minimum level of individual rights everywhere. State constitutions can promise more, they just can't promise less.
The federal Constitution is the floor
Through the Fourteenth Amendment, most of the Bill of Rights has been 'incorporated' against the states, setting a nationwide minimum: no state may drop below the federal guarantee of free speech, due process, protection from unreasonable searches, and the rest.
State constitutions can raise the ceiling
A state is free to protect rights more generously than the federal floor under its own constitution, even using identical words. In PruneYard Shopping Center v. Robins (1980), the Supreme Court upheld California's grant of broader free-expression rights in a private shopping center than the First Amendment requires. Several states recognize privacy or search-and-seizure protections that exceed the federal baseline.
Why 'same words, higher ceiling' is possible
A state high court interpreting its own constitution is the final authority on that text (see Supremacy). So it may read its state due-process or search clause to give more protection than the Supreme Court reads into the parallel federal clause, without any conflict, because it is adding protection, not subtracting it.
The limit
The ceiling only goes up. A state cannot use its constitution to fall below the federal floor for individual rights; where it tries, the federal minimum controls through the Supremacy Clause.