Andy Worthington, author of the Guantanamo Files: The Stories of the 774 Detainees in America's Illegal Prison, unearths details about the two men challenging the US's right to detain them, in a case at the Supreme Court.
The cases were filed by US lawyers on behalf of Kuwaiti Fawzi al-Odah and 11 other Kuwaiti detainees, and Algerian-born Bosnian Lakhdar Boumediene and five other Bosnian detainees.
At issue is the 2006 Military Commissions Act, passed by Congress and signed by President George W Bush.
Fawzi al-Odah is one of three Kuwaitis still held in Guantanamo
It stripped Guantanamo Bay detainees of the right to challenge in federal courts the basis of their indefinite detention as "enemy combatants".
Their cases could only be heard in military commissions, not civilian courts, it stated.
Lawyers for the detainees say this violates their constitutional right to habeas corpus - a procedure under which someone who holds a prisoner is required to explain to a court the reason why.
The administration disagrees, arguing that habeas corpus does not apply to non-US sovereign territory.
The main sources of information about both men are the accounts they have given to US military tribunals. If you ask officials at Guantanamo Bay for information, they will point you towards these documents.
Fawzi al-Odah, 30, is a Kuwaiti primary school teacher whose father, a retired air force pilot, fought with US forces during the Gulf War in 1991.
According to his own account, which he gave to a military tribunal at Guantanamo, he took a short holiday from work and travelled to Afghanistan in August 2001 to teach the Koran and provide humanitarian aid.
Occupation: School teacher
This was something he had done before, in other countries, and his family had had a history of providing humanitarian aid, establishing libraries and wells in various countries in Africa.
After establishing contact with the Taleban, which he said "was necessary because that was the government in Afghanistan at that time", Mr Odah said he had been "touring the schools and visiting families", teaching the Koran and handing out money, until his activities had been curtailed following 9/11.
He said that in Kandahar the Taleban representative "told me that was a dangerous place because it was the capital for the Taleban", and had advised him to go to Logar, in the east of the country, where he had stayed with a family for a month, and left his passport and belongings for safekeeping.
"If the Afghans saw I had a passport indicating I was an Arab, and they saw the money and the camera I had, I would have been killed," he added.
He had then moved to Jalalabad, where he had stayed with another family, who had given him an AK-47 assault rifle to protect himself, Mr Odah said.
He had then joined other people crossing the mountains to Pakistan, where he had handed himself in to the border guards, he added.
Mr Odah said he expected to be escorted to the Kuwaiti embassy, but had instead been handed over to US forces.
He joined a widespread hunger strike in Guantanamo in August 2005.
In his latest military review, he was not accused of participating in hostilities against US forces, but of:
- "firing a Kalashnikov [AK-47] rifle at some targets" at a small camp where he had been taken by a Taleban official
- staying in a house in Jalalabad "with three Arabs who appear to be fighters who carried Kalashnikovs", and
- fleeing Afghanistan with a group of men "who may have had some al-Qaeda or Taleban members"
Eight of the Kuwaiti detainees have already been released from Guantanamo.
According to Mr Odah's lawyers, the State Department's senior legal adviser, John Bellinger, said Mr Odah and the three other Kuwaiti detainees were still in Guantanamo because the Kuwaiti government had broken the rules the US administration had tried to impose on them when the other eight men had been released.
The lawyers say the Americans had wanted the men to be tried and imprisoned on their return, but the Kuwaiti court hearing the trial found no case to answer and released them.
Lakhdar Boumediene, now 41, travelled to Bosnia with five other Algerian men during the civil war in the 1990s, and may have fought with Bosnian forces against the Serbs.
The six stayed in Bosnia, married Bosnian women, were granted citizenship and took jobs working with orphans for various Muslim charities.
In October 2001, the US embassy in Sarajevo asked the Bosnian government to arrest them because of a suspicion they had been involved in a plot to bomb the embassy.
Occupation: Charity worker
The six men were duly arrested. But after a three-month investigation, in which the Bosnian police searched their apartments, their computers and their documents, there was - according to a report by the New-York-based Center for Constitutional Rights - still no evidence to justify the arrests.
Bosnia's Supreme Court ordered their release, and the Bosnian Human Rights Chamber ruled they had the right to remain in the country and were not to be deported.
However, on the night of 17 January 2002, after they were freed from Bosnian custody, they were seized and rendered to Guantanamo.
Since arriving in Guantanamo, the men have faced repeated allegations of links to al-Qaeda - but the embassy plot has never been mentioned.
It was alleged in a tribunal hearing that an unidentified source had said Mr Boumediene "was known to be one of the closest associates of an al-Qaeda member in Europe".
The men have persistently denied the allegations.
Their lawyers say the source of the bomb-plot allegations was the embittered former brother-in-law of one of the men, who ran a smear campaign against him.
The UN special rapporteur on torture, Manfred Novak, has said: "It's implausible to say that they are enemy combatants.
"They were fighters during the Bosnian war, but that ended in 1995.
"They may be radical Islamists, but they have definitely not committed any crime."
According to the Washington Post, they were formally exonerated by Bosnian prosecutors in 2004.
Families of the Algerians seized in Bosnia have protested in Sarajevo
In March 2005, US Secretary of State Condoleezza Rice responded to a request for their release from the Bosnian prime minister by saying it was not possible because "they still possess important intelligence data".
All six men have said they have been treated brutally in Guantanamo, subjected to "enhanced interrogation techniques" involving prolonged isolation, forced nudity and sleep deprivation.
The lawyers of one of the six, 37-year-old Mustafa Ait Idir, have backed up allegations he had a stroke after an assault by guards in Guantanamo, which left one side of his face paralysed.
Andy Worthington is the author of The Guantanamo Files: The Stories of the 774 Detainees in America's Illegal Prison (Pluto Press/University of Michigan Press, 2007).