Can calculations cause arithmetic overflow even if final value fits? [migrated]

I am reading Algorithm Design Manual by Skiena where in Chapter 8, Section 8.1.4 where it talks about calculation of binomial coefficients, it says:

 "Intermediate calculations can easily cause arithmetic overflow,
even when the final coefficient fits comfortably within an integer."

I can't think of a reason why. Shouldn't whatever calculation is done in between always be less than the final one?