We present deep near-infrared Ks-band Anglo-Australian Telescope Infrared Imager and Spectrograph observations of a selected sample of nearby barred spiral galaxies, including some with the strongest known bars. The sample covers a range of Hubble types from SB0(-) to SBc. The goal is to determine if the torque strengths of the spirals correlate with those of the bars, which might be expected if the bars actually drive the spirals as has been predicted by theoretical studies. This issue has implications for interpreting bar and spiral fractions at high redshift. Analysis of previous samples suggested that such a correlation exists in the near-infrared, where effects of extinction and star formation are less important. However, the earlier samples had only a few excessively strong bars. Our new sample largely confirms our previous studies, but still any correlation is relatively weak. We find two galaxies, NGC 7513 and UGC 10862, where there is only a weak spiral in the presence of a very strong bar. We suggest that some spirals probably are driven by their bars at the same pattern speed, but that this may be only when the bar is growing or if there is abundant gas and dissipation.