  In this work, we investigate the reliability of the BPT diagram for excluding
galaxies that host an AGN. We determine the prevalence of X-ray AGN in the
star-forming region of the BPT diagram and discuss the reasons behind this
apparent misclassification, focusing primarily on relatively massive
($\log(M_{*})\gtrsim10$) galaxies. X-ray AGN are selected from deep XMM
observations using a new method that results in greater samples with a wider
range of X-ray luminosities, complete to $\log(L_{X})>41$ for $z<0.3$. Taking
X-ray detectability into account, we find the average fraction of X-ray AGN in
the BPT star-forming branch is 2$\%$, suggesting the BPT diagram can provide a
reasonably clean sample of star-forming galaxies. However, the X-ray selection
is itself rather incomplete. At the tip of the AGN branch of the BPT diagram,
the X-ray AGN fraction is only 14$\%$, which may have implications for studies
that exclude AGN based only on X-ray observations. Interestingly, the X-ray AGN
fractions are similar for Seyfert and LINER populations, consistent with LINERs
being true AGN. We find that neither the star-formation dilution nor the hidden
broad-line components can satisfactorily explain the apparent misclassification
of X-ray AGN. On the other hand, $\sim40\%$ of all X-ray AGN have weak emission
lines such that they cannot be placed on the BPT diagram at all and often have
low specific SFRs. Therefore, the most likely explanation for "misclassified"
X-ray AGN is that they have intrinsically weak AGN lines, and are only
placeable on the BPT diagram when they tend to have high specific SFRs.
