We study whether binary black hole template banks can be used to search for the gravitational waves emitted by general binary coalescences. To recover binary signals from noisy data, matched-filtering techniques are typically required. This is especially true for low-mass systems, with total mass M less than or similar to 10 M-circle dot, which can inspiral in the LIGO and Virgo frequency bands for thousands of cycles. In this paper, we focus on the detectability of lowmass binary systems whose individual components can have large spin-induced quadrupole moments and small compactness. The quadrupole contributes to the phase evolution of the waveform whereas the compactness affects the merger frequency of the binary. We find that binary black hole templates (with dimensionless quadrupole k = 1) cannot be reliably used to search for objects with large quadrupoles (k greater than or similar to 20) over a wide range of parameter space. This is especially true if the general object is highly spinning and has a larger mass than its binary companion. A binary that consists of objects with small compactness could merge in the LIGO and Virgo frequency bands, thereby reducing its accumulated signal-to-noise ratio during the inspiraling regime. Template banks which include these more general waveforms must therefore be constructed. These extended banks would allow us to realistically search for the existence of new astrophysical and beyond the Standard Model compact objects.