Solution to the Problem:

The number is 91.
Originally, I thought the correct answer was 1729 because I did not consider cubes of negative numbers. Richard Johnson was the first to send in the solution of 91.
I decided to allow both answers, 91 and 1729.
The numbers that you cube to get them are:
-5³ + 6³ = 91; and 3³ + 4³ = 91 and
1³ + 12³ = 1729; and 9³ + 10³ = 1729.

Keith Mealy sent in the following:
There is a famous story of the English mathematician, Hardy, visiting the Indian mathematical genius, Ramanujan, in hospital. Hardy remarked that the taxi he had come in had the number 1729, which struck him as not a very interesting number. Ramanujan disagreed, pointing out that it is the smallest positive integer that can be written in two different ways as the sum of two cubes (of positive whole numbers).