Playing in Excel a bit with the cubes of 1 through 10,000, the maximum and only workable value I've found for N (so far) is 181; that is:
105^3 - 104^3 = 1,157,625 - 1,124,864 = 32,761 = 181^2, with 2*181 + 79 = 441 = 21^2.
Not sure whether 181 is necessarily 'the maximum value'. Would like to explore further beyond 10,000, but the 2010 Winter Olympic Torch has just arrived here in Toronto and the relay's arriving outside my office building for a big ceremony in just a little while. Can't miss that!!