Harmonic errors in acoustic instruments
A few things:


Not all instruments have partials neatly defined by the $h_n = nf$. Kettle drums would be an obvious example, but bells are another. In this case, you will be way off looking in the "usual places" +/- 5%.
Many instruments have only even or odd harmonics. Others are designed to omit some harmonics. The piano, for example, is designed to have almost no 7th harmonic. This may not matter to you, but it's worth pointing out in case it does.
It is common for higher harmonics to be "out of tune", that is, not in the "usual places". An engineer who once worked as a piano tuner once admonished me for using harmonics to tune my upright bass (a technique that works well for lighter-stringed instruments), because, he said, the partials are more out of tune with heavier stringed instruments. He said it's especially true told of higher harmonics, explaining that the math that gives you the simple formula for partials assumes that strings have no mass. 5% strikes me as a little much (almost a semi-tone!), but I don't think it's out of the question.
Many instruments have vibrato. This throws all your harmonics out of wack: