For p in [2,infty ), we consider the L^p rightarrow L^p boundedness of a Nikodym maximal function associated to a one-parameter family of tubes in {mathbb {R}}^{d+1} whose directions are determined by a non-degenerate curve gamma in {mathbb {R}}^d. These operators arise in the analysis of maximal averages over space curves. The main theorem generalises the known results for d = 2 and d = 3 to general dimensions. The key ingredient is an induction scheme motivated by recent work of Ko-Lee-Oh.