For example, in music we use numbers for the notes of a scale (G is the 5th of C, because in the C major scale: C=1, D=2, E=3, F=4, G=5, A=6, B=7). The numbers are clearly indices for the notes of the scale.

But we often think of stuff like: what's the 3rd of the 5th -- that is, get the note at the 5th position (G in our example) and then get the 3rd of that note (the 3rd of G is B). But note that B is the 7th of C major, not the 8th you'd get from adding 5 and 3.

The problem, of course, is that we ended up using the numbers as offsets, even though they started as indices.