The fall-out shelter, deep underground at the former prisoner of war camp at Cultybraggan, near Comrie, Perthshire, was designed to be impenetrable, but it was stripped of £30,000 of copper by the criminals.
The thieves are believed to have ripped out showers, sinks and other equipment in raids spanning from June to earlier this month, when the owners noticed.
Police are investigating the break-ins. A spokeswoman said: "It's understood damage caused to the premises could run into six figures."
It comes after thieves last week stole copper from an electricity substation in Alloa, Clackmannanshire, and earlier in the summer more than a mile of copper train signalling cable was taken in Aberdeenshire.
The Cultybraggan Camp was set up in 1939 as a maximum security prison and housed up to 4000 German and Italian PoWs, with Adolf Hitler's deputy, Rudolf Hess, said to have been among them.
The windowless former Government bunker was designed to accommodate up to 150 key officials in the event of a nuclear attack.
It was kitted out with a TV studio, canteen, telephone exchange and dormitories, decontamination showers, a public announcement system and a radio mast.
But although it cost the nation £30 million when the threat was felt to be very real, it was not completed until 1990, just as the Cold War was ending.
It is thought to be the last and most technologically advanced structure built specifically in relation to the Cold War threat.
The two-level bunker has 27 rooms above ground level and 22 below.
In 2007, Cultybraggan was bought from the Ministry of Defence for £150,000 through a community right-to-buy option and since then work has gone into securing its future. The trust appointed property consultants to sell the bunker and it was put on the market with a guide price of £400,000.