Adding to the mystery, the Canadian embassy has reported that some of its staff became ill. Canada and Cuba have long been more friendly relations than relations between the United States and Cuba.
The government realized in January that something had happened, but it took months of tests to identify the 24 people affected.
Following the attacks, President Trump expelled 15 Cuban diplomats from the United States, the most serious of a series of actions he has undertaken to resolve the detente between the countries undertaken by President Barack Obama.
Mr. Obama sought to put an end to the hostility and distrust that had characterized the relationship of the countries for more than half a century, and considered the resumption of relations as one of his foreign policy legacies.
The State Department said that the expulsion of the diplomats was intended to force Cuba to place its embassy in Washington, where the diplomats were stationed, in the same state of emergency that the United States is now operating in Havana after he decided to reduce his staff to a skeletal group of only 27 people.
Foreign Minister Bruno Rodríguez de Cuba said in November that the expulsions were unjustified and he complained that the United States had not cooperated with the Cuban authorities in his investigation. In his first response to Mr. Rodriguez, Mr. Tillerson said he would not allow officials to share private information with Cubans or to provide "whoever is orchestrating these attacks with useful information to know how effective they were."
"We understand Cubans do not like the actions we have taken," added Mr. Tillerson. "We do not like our diplomats being attacked."
Responding to a question about the plans of John F. Kelly, White House chief of staff, to replace Mr. Tillerson with Mike Pompeo, C.I.A. director, Mr. Tillerson denied the accuracy of the reports.
"This is a story that continues to appear every six weeks," said Mr. Tillerson. "I would say that everyone needs to get new sources because their story is still incorrect"
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