Cockroaches are capable of surviving higher levels of radiation than humans for two main reasons. They have a tough exoskeleton which blocks some of the most deadly radiation and they can hide away in much smaller spaces. Rumours about the ability of a cockroach to survive a nuclear bomb are not true – this is not because of radiation but because of the heat produced by the bomb. Temperatures can rise to around 10 million celsius – not even a cockroach could survive that.
As Michael says, no living thing can survive a nuclear blast. If you were far enough away from a nuclear bomb you could survive the blast – but the radiation would make you very sick and eventually kill you if the dose was high enough. As Michael also says, cockroaches can survive higher levels of radiation than humans.