We compute the breakdown point of the subsampling quantile of a general statistic, and show that it is increasing in the subsampling block size and the breakdown point of the statistic. These results imply fragile subsampling quantiles for moderate block sizes, also when subsampling procedures are applied to robust statistics. This instability is inherited by data driven block size selection procedures based on the minimum confidence interval volatility (MCIV) index. To overcome these problems, we propose for the linear regression setting a robust subsampling method, which implies a sufficiently high breakdown point and is consistent under standard conditions. Monte Carlo simulations and sensitivity analysis in the linear regression setting show that the robust subsampling with block size selection based on the MCIV index outperforms the subsampling, the classical bootstrap and the robust bootstrap, in terms of accuracy and robustness. These results show that robustness is a key aspect in selecting data driven subsampling block sizes.