  Recent observations by pulsar timing arrays indicate the presence of
gravitational wave signals, likely from supermassive black hole binaries. These
binaries can produce two types of signals: a stochastic gravitational wave
background (GWB) or deterministic continuous gravitational waves (CGW).
Correctly separating these signals is essential for accurate signal recovery
and interpretation. This paper investigates the interaction between GWB and CGW
signals within current analysis pipelines, focusing on misinterpretations and
biases in parameter estimation when these signals are analyzed separately or
together. We conducted several simulations based on the European PTA 24.8-year
dataset. First, we injected either a GWB or a CGW. We compared parameter
estimation using different search models, including Earth term only, combined
Earth and pulsar term CGW templates, and correlated (HD) or uncorrelated (CURN)
power law GWB templates. Our findings show that, when searched independently,
the GWB and CGW signals can be misinterpreted as each other, and only a
combined search can recover the true signal present. For datasets containing
both a GWB and a CGW, failure to account for the CGW biases the recovery of the
GWB. However, a combined search allows for the recovery of both GWB and CGW
parameters without significant bias. Care must be taken with the method used to
perform combined searches on these multi-component datasets, as the CGW pulsar
term can be misinterpreted as a CURN. This can be avoided by direct searches
for an HD correlated GWB plus a CGW. Our study underscores the importance of
combined searches to ensure unbiased recovery of GWB parameters in the presence
of strong CGWs, which is crucial for interpreting signals recently found in PTA
data and represents a step toward a robust framework for disentangling
stochastic and deterministic GW signals in future, more sensitive datasets.
