Yes, up to a week. Cockroaches are said to be one of the most tenacious creatures on earth and are often cited as the most likely survivors of a nuclear war. They can easily adapt to, and thus survive, harsh conditions.
Another strange fact about the cockroach is that it can survive without its head for up to a week. While humans and other animals are considered worm food the instance their heads are chopped off, a headless cockroach will continue to live much longer than other decapitated creatures.
This is because cockroaches have unique nervous and respiratory systems which means that they can breathe even without their heads. They also do not bleed to death when their heads are chopped off because they don't have a huge network of blood vessels like that of humans and other animals. Any wound is immediately sealed off by clotting. In fact, cockroaches do not need blood to supply oxygen to every part of its body. They breathe through spiracles which deliver air to each cell of the body through a set of tubes called tracheae.
Cockroaches also have very low metabolism which means that their last meal can sustain them for a long time especially since they don't move around pretty much after decapitation.
Problems starts when dehydration or starvation sets in because they cannot feed. Infection from mold, bacterium or virus, can also contribute to the headless cockroach's premature death.