We investigate the use of subsampling for conducting inference about the quadratic variation of a discretely observed diffusion process under an infill asymptotic scheme. We show that the usual subsampling method of Politis and Romano (1994) is inconsistent when applied to our inference question. Recently, a type of subsampling has been used to do an additive bias correction to obtain a consistent estimator of the quadratic variation of a diffusion process subject to measurement error, Zhang, Mykland, and Ait-Sahalia (2005). This subsampling scheme is also inconsistent when applied to the inference question above. This is due to a high correlation between estimators on different subsamples. We discuss an alternative approach that does not have this correlation problem; however, it has a vanishing bias only under smoothness assumptions on the volatility path. Finally, we propose a subsampling scheme that delivers consistent inference without any smoothness assumptions on the volatility path. This is a general method and can be potentially applied to conduct inference for quadratic variation in the presence of jumps and/or microstructure noise by subsampling appropriate consistent estimators.