The BBC said the handwritten letter, mailed March 3, 1950, was addressed to George Green at the university's Trinity College.
It reads: "George, will meet at Monty's next weekend. Is 2 p.m. acceptable? Love Gwen."
The college is checking its records to see if Green was a student.
A Royal Mail spokesman said the fact that the letter has a postmark makes is highly unlikely that it has been in their system for more than 50 years.
"It is possible that it may have got caught up in a large envelope and sent to a wrong address," spokesman James Taylor told the BBC. "It is certainly a strange one."