  The eccentricity of binary black-hole mergers is predicted to be an indicator
of the history of their formation. In particular, eccentricity is a strong
signature of dynamical formation rather than formation by stellar evolution in
isolated stellar systems. We investigate the efficacy of the existing
quasicircular parameter estimation pipelines to determine the source parameters
of such eccentric systems. We create a set of simulated signals with
eccentricity up to 0.3 and find that as the eccentricity increases, the
recovered mass parameters are consistent with those of a binary with up to a
$\simeq 10\%$ higher chirp mass and mass ratio closer to unity. We also employ
a full inspiral-merger-ringdown waveform model to perform parameter estimation
on two gravitational wave events, GW151226 and GW170608, to investigate this
bias on real data. We find that the correlation between the masses and
eccentricity persists in real data, but that there is also a correlation
between the measured eccentricity and effective spin. In particular, using a
nonspinning prior results in a spurious eccentricity measurement for GW151226
as it exhibits signs of nonzero black-hole spin. Performing parameter
estimation with an aligned spin, eccentric model, we constrain the
eccentricities of GW151226 and GW170608 to be $<0.15$ and $<0.12$ respectively.
