We present upper limits on the 21 cm power spectrum at $z = 5.9$ calculated from the model-independent limit on the neutral fraction of the intergalactic medium of $x_{rm H{small I }} < 0.06 + 0.05 (1sigma)$ derived from dark pixel statistics of quasar absorption spectra. Using 21CMMC, a Markov chain Monte Carlo Epoch of Reionization analysis code, we explore the probability distribution of 21 cm power spectra consistent with this constraint on the neutral fraction. We present 99 per cent confidence upper limits of $Delta^2(k) < 10$ to $20 {rm mK}^2$ over a range of $k$ from 0.5 to $2.0 h{rm Mpc}^{-1}$, with the exact limit dependent on the sampled $k$ mode. This limit can be used as a null test for 21 cm experiments: a detection of power at $z=5.9$ in excess of this value is highly suggestive of residual foreground contamination or other systematic errors affecting the analysis.
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