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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.