  Waveform templates are a powerful tool for extracting and characterizing
gravitational wave signals, acting as highly restrictive priors on the signal
morphologies that allow us to extract weak events buried deep in the
instrumental noise. The templates map the waveform shapes to physical
parameters, thus allowing us to produce posterior probability distributions for
these parameters. However, there are attendant dangers in using highly
restrictive signal priors. If strong field gravity is not accurately described
by General Relativity (GR), then using GR templates may result in fundamental
bias in the recovered parameters, or even worse, a complete failure to detect
signals. Here we study such dangers, concentrating on three distinct
possibilities. First, we show that there exist modified theories compatible
with all existing tests that would fail to be detected by the LIGO/Virgo
network using searches based on GR templates, but which would be detected using
a one parameter post-Einsteinian extension. Second, we study modified theories
that produce departures from GR that turn on suddenly at a critical frequency,
producing waveforms that do not naively fit into the simplest parameterized
post-Einsteinian (ppE) scheme. We show that even the simplest ppE templates are
still capable of picking up these strange signals and diagnosing a departure
from GR. Third, we study whether using inspiral-only ppE waveforms for signals
that include merger and ringdown can lead to problems in misidentifying a GR
departure. We present an easy technique that allows us to self-consistently
identify the inspiral portion of the signal, and thus remove these potential
biases, allowing GR tests to be performed on higher mass signals that merge
within the detector band. We close by studying a parameterized waveform model
that may allow us to test GR using the full inspiral-merger-ringdown signal.
