Over 1,000 ex-serviceman claim they became ill after being exposed to radiation during nuclear weapons tests by the Ministry of Defence in the 1950s.
David Harman, who served in the RAF on Christmas Island in 1957 and 1958, described the way in which the MoD had treated veterans as "shabby".
During the nuclear explosions, he was only told to turn his back, close his eyes and cover them with his fists. He and his colleagues later suffered from mystery illnesses.
"The trouble is really that, particurlaly at the time, the medical profession just stood there and said: 'we have no idea what's caused that', Mr Harman said.
More than 1,000 veterans want to claim damages from the Ministry of Defence. Last year, the Court of Appeal ruled against them after hearing arguments in a series of "test cases". Appeal judges said in November 2010 that veterans could not claim damages because they had no evidence by which they could hope to prove that illnesses had probably been caused by radiation exposure.
Lawyers representing the veterans asked three Supreme Court justices for permission to argue their right to seek damages in the Supreme Court. Lord Phillips, Lady Hale and Lord Brown gave permission.
The Supreme Court is expected to hear veterans' arguments for the right to go ahead with damages claims either later this year or next year. If the Supreme Court was to rule that claims could go ahead, veterans would be likely to launch damages claims in the High Court.
Announcing the decision, Lord Phillips, president of the Supreme Court, said: "The application for permission to appeal is granted, but may I just emphasise that this is only an application for permission to appeal, and the court would not wish to raise false optimism in what are obviously very difficult cases.