  Detections of gravitational waves emitted by binary black holes allow for
tests of General Relativity in the strong-field regime. In particular,
deviations from General Relativity can be observed by comparing incoming
signals to waveform templates that include parametrized deviations from General
Relativity. However, it is essential that the General Relativity sector of
these templates accounts for all predictable physics. Otherwise, missing
physics might be mimicked by the "beyond General Relativity" sector of the
templates, leading the analysis to report apparent deviations from General
Relativity. Current parametrized tests implement templates that omit physical
phenomena such as orbital eccentricity and higher-order modes. In this paper,
we show how the omission of higher modes can lead to false deviations from
General Relativity when these effects are strong enough. We study the extent of
these deviations as a function of the mass ratio and the orbital orientation.
We find that significant false deviations can arise when current tests are
performed on signals emitted by asymmetric binaries whose orbital angular
momentum is orthogonal to the line-of-sight. We estimate that the Advanced
LIGO-Virgo network operating at its design sensitivity can observe false
violations with a significance above $5 \sigma$ as often as once per year.
Similar results are expected for other tests of General Relativity that use
modified waveform models. Hence, we stress the necessity of accurate waveform
models that include physical effects such as higher-order modes to trust future
tests of General Relativity.
