If I understand hinkley's point correctly, it's the constant multiplier (and, in some cases, lower-order terms) that conventional complexity analysis handwaves away as irrelevant because it only looks at the limit as input size goes to infinity.

I guess the op meant the C as in

Total time = nX + C

No matter how much you optimize X (n here is for instance the number of items), if C is significant then it matters. It is usually harder to optimize the constant C in the OP's experience.