An Internal Security Agency official said Tuesday's messages were signed anon.pl, Polskie Radio reported.
"There is no certainty that it was actually the Polish branch of Anonymous which sent the emails," he told TVN24. "We do not rule out that someone is impersonating them."
Anonymous hacked into government websites last year, saying it was a reprisal for Poland's signing a treaty on Internet piracy. This week's hoax emails said they were being sent because "someone is targeting us," the news site Dziennik.pl reported.
More than 2,000 people were evacuated from hospitals, prosecutor's offices and other institutions in Warsaw and elsewhere because of the hoax.
Officials said one man has been detained in Chrzanow in southern Poland where a hospital was targeted.
Anyone convicted of sending the threats could be sentenced to a possible eight years in prison.