Renowned cellist Lynn Harrell travels the world on two airline tickets—one for himself and one for his beloved companion, his $5 million, nearly 300-year-old cello. "Cello Harrell" gets its own boarding pass and even collects frequent-flier miles in its own account. Many cellos do, even for high school musicians.

But two seats don't always equal twice the miles. Delta Air Lines' frequent-flier rules only allow people, not musical instruments, to collect miles, and Delta has been auditing accounts to catch frequent-flier...