GWALIOR, INDIA—Nobody knows exactly when or why the witnesses and small-time crooks caught up in one of India’s biggest-ever corruption scandals began dying under mysterious circumstances. But in the past two years, that’s what has happened to more than two dozen people implicated in a $1 billion test-rigging scheme.

Even by standards in India, where corruption is routine, the scale of the scam in the central state of Madhya Pradesh is mind-boggling. Police say that since 2007, tens of thousands of students and job aspirants have paid hefty bribes to middlemen, bureaucrats and politicians to rig test results for medical schools and government jobs.

So far, 1,930 people have been arrested and more than 500 are on the run. Hundreds of medical students are in prison — along with several bureaucrats and the state’s education minister. Even the governor has been implicated.

Police have had their hands full racing to meet a July deadline in the criminal probe. And now they are faced with the deaths of more witnesses and suspects. Within the last month, police said, one of those accused died after having chest pains in prison, another drowned in a village pond and a third died of a liver infection.

On July 4, television reporter Akshay Singh died while investigating a suspect’s death. Singh sipped tea during an interview and began coughing and foaming at the mouth, according to media reports. He was rushed to the hospital, where doctors said he had suffered a heart attack. Police said the initial examination did not reveal anything “suspicious.”

The state’s government, run by the Bharatiya Janata Party, has said that “no conspiracy was found” in the recent deaths. But others involved in the case fear otherwise. The state’s chief minister, Shivraj Singh Chouhan, said recently that his mind is in “agony and pain” and promised that all the deaths will be investigated.

“The police say they keep coming up against a wall in their investigation every time someone is found dead,” said Chandresh Bhushan, chairman of the special investigation team that was appointed by the state court to monitor the police probe. “We ask them, ‘Why are so many dying in road accidents in this case? Does this have any link to the scam?’ There is no evidence of a link yet, but we cannot overrule it, either.”

But in Madhya Pradesh, cheating on tests became a sophisticated racket using various approaches. High-scoring students from around the country toting fake identity cards were brought in to impersonate applicants taking tests. Applicants would be told to leave their answer sheets blank so that scorers could fill them out in a separate room. And scores would be manipulated by testing board officials to favour bribe-paying candidates.

In July 2013, after a tip-off from whistleblowers, police raided a test centre in Indore and arrested eight impersonators taking the medical school test. Police say that the scheme was operated by a syndicate of agents, doctors, officials and politicians across five states and that there was no single ringleader.

“There is so much information with the investigators that it could bring the government down,” said Ashish Chaturvedi, 26, one of the whistleblowers. He has been attacked 14 times by unknown assailants, he said. Six of the assaults took place in front of a police officer assigned by a court to protect him last year.

“But so far they have just gone after students, middlemen and lower officials,” Chaturvedi said. “They are not even probing the roles of the big fish in the state.”

The news portal the Wire called the scam “one of the most complex, multi-layered scandals affecting public life in India,” one that “is fast acquiring the shades of a macabre crime thriller.”

Many of the mysteriously dead are young — either students who paid money and made it into medical schools or job aspirants trying to take tests to become police officers, school teachers, forest rangers and food inspectors.