  Resonator measurements are a simple but powerful tool to characterize a
material's microwave response. The losses of a resonant mode are quantified by
its internal quality factor $Q_\mathrm{i}$, which can be extracted from the
scattering coefficient in a microwave reflection or transmission measurement.
Here we show that a systematic error on $Q_\mathrm{i}$ arises from Fano
interference of the signal with a background path. Limited knowledge of the
interfering paths in a given setup translates into a range of uncertainty for
$Q_\mathrm{i}$, which increases with the coupling coefficient. We
experimentally illustrate the relevance of Fano interference in typical
microwave resonator measurements and the associated pitfalls encountered in
extracting $Q_\mathrm{i}$. On the other hand, we also show how to characterize
and utilize the Fano interference to eliminate the systematic error.
