  We analyze the microlensing event OGLE-2019-BLG-0304, whose light curve
exhibits two distinctive features: a deviation in the peak region and a second
bump appearing $\sim 61$~days after the main peak. Although a binary-lens model
can explain the overall features, it leaves subtle but noticeable residuals in
the peak region. We find that the residuals can be explained by the presence of
either a planetary companion located close to the primary of the binary lens
(3L1S model) or an additional close companion to the source (2L2S model).
Although the 3L1S model is favored over the 2L2S model, with $\Delta\chi^2\sim
8$, securely resolving the degeneracy between the two models is difficult with
the currently available photometric data. According to the 3L1S interpretation,
the lens is a planetary system, in which a planet with a mass
$0.51^{+0.51}_{-0.23}~M_{\rm J}$ is in an S-type orbit around a binary composed
of stars with masses $0.27^{+0.27}_{-0.12}~M_\odot$ and
$0.10^{+0.10}_{-0.04}~M_\odot$. According to the 2L2S interpretation, on the
other hand, the source is composed of G- and K-type giant stars, and the lens
is composed of a low-mass M dwarf and a brown dwarf with masses
$0.12^{+0.12}_{-0.05}~M_\odot$ and $0.045^{+0.045}_{-.019}~M_\odot$,
respectively. The event illustrates the need for through model testing in the
interpretation of lensing events with complex features in light curves.
