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Our view of the last-scattering surface in the cosmic microwave background (CMB) is obscured by secondary anisotropies, sourced by scattering, extragalactic emission and gravitational processes between recombination and observation. Whilst it is established that non-Gaussianity from the correlation between the integrated-Sachs-Wolfe (ISW) effect and gravitational lensing can significantly bias primordial non-Gaussianity (PNG) searches, recent work by Hill (2018) has suggested that other combinations of secondary anisotropies can also produce significant biases. Building on that work, we use the WebSky and Sehgal et al.(2010) simulations to perform an extensive examination of possible biases to PNG measurements for the local, equilateral and orthogonal shapes. For a Planck-like CMB experiment, without foreground cleaning, we find significant biases from cosmic infrared background (CIB)-lensing and thermal Sunyaev-Zel'dovich (tSZ)-lensing bispectra for the local and orthogonal templates, and from CIB and tSZ bispectra for the equilateral template. For future experiments, such as the Simons Observatory, biases from correlations between the ISW effect and the tSZ and CIB will also become important. Finally we investigate whether foreground-cleaning techniques are able to suppress these biases sufficiently. We find that the majority of these biases are effectively suppressed by the internal-linear-combination method and the total bias to Planck-like and SO-like experiments is less than the $1\,σ$ statistical error. However, the small total bias arises from the cancellation of several $1\,σ$ biases for Planck-like experiments and $2\,σ$ biases for SO-like. As this cancellation is likely sensitive to the precise modelling, to ensure robustness against these biases explicit removal methods should be used, likely at the cost of decreased constraining power.