This paper is one of the first to investigate mobility in overall health using high‐quality administrative data. The attractiveness of this approach lies in objective health measures and large sample sizes allowing twin analyses. I operationalize health mobility by a variety of statistics: rank–rank slopes, intergenerational correlations (IGCs) and sibling and identical twin correlations. I find rank–rank slopes and IGCs in the range 0.11–0.15 and sibling correlations in the range 0.14–0.20. Mobility in health is thus relatively high, both when compared to similar US‐based studies, and when contrasted with outcomes such as educational attainment and income. Comparing sibling and identical twin correlations with parent–child associations confirms earlier findings in the literature on equality of opportunity, namely that sibling correlations capture far more variation than traditional IGCs. I conclude that 14%–38% of the variation in individual health outcomes can be attributed to family background and genes, factors which the individual cannot be held accountable for. This finding suggests that simple parent–child associations may be a poor metric for measuring health mobility.