According to the documents, the response from Nike came 10 hours later. “Got it,” Mr. Lotwis responded. “I will submit right away. Thanks.”

Within days, according to bank records, the $500,000 was withdrawn by Athletics Kenya’s top officials. There were no major track and field activities going on at the time, and the board member and the former administrative assistant said just about all of the money had been concealed from Athletics Kenya’s executive committee, including $200,000 sent to a bank account in Hong Kong. Several analysts said the chairman’s asking for the money to be wired to his personal account and then sending a follow-up email labeled “Urgent!!” should have been a tipoff to Nike that something was untoward.

Mr. Kiplagat and two of the other implicated officials have denied any wrongdoing.

A federal prosecutor consulted for his opinion on these allegations said that in many corruption cases, concealment was an indicator of criminal intent and that in this case, there was no obvious concealment by Nike — the commitment bonus was explicitly laid out in the contract. The prosecutor, who spoke on the condition he would not be identified, also said it would be very difficult to prove that Nike executives knew that the Kenyan officials were going to steal the bonus.

The former administrator, however, has little doubt.

“Because of Nike’s secretive meetings with the top three A.K. officials,” he said in his affidavit to investigators, “it is my opinion that Nike officials have always been aware that the payments in question are improper.”

Analysts said this case was especially tricky because it did not appear to fall under the Foreign Corrupt Practices Act, the American law that covers crimes involving American companies and foreign government officials. The Kenyan running association, while it receives some government money, is not a Kenyan government agency. “The world of sports has evolved in a governance hole,” said Roger Pielke Jr., a political scientist at the University of Colorado.

He noted that sports federations, like Athletics Kenya and FIFA, international soccer’s governing body, which is embroiled in its own corruption saga, often fell between the cracks of the rules that governed businesses, public agencies and traditional nonprofit organizations, even though sports federations have qualities of all three.

Mr. Pielke said bribery, embezzlement and “unsavory, improper business practices” were common.

“I hear it all the time from sporting officials,” he said. “ ‘To survive in this world, these are the rules of the game.’ ”