Ordinal sums of triangular norms (t-norms) on bounded lattices are studied. Originally, t-norms were defined on the unit interval. Later, a generalization on a more general algebraic structure, namely bounded lattices, was proposed. It turned out that in this case, an ordinal sum in Clifford's sense of t-norms may not be a t-norm. Hence, it is important to state necessary and sufficient conditions for ensuring whether an ordinal sum of arbitrary t-norms on a bounded lattice is a t-norm, too. In this paper, several conditions are formulated in order to verify whether an ordinal sum of a particular family of t-norms is a t-norm. First, ordinal sums with one t-norm are studied. The general case of an ordinal sum of a family of t-norms on pairwise disjoint subintervals is also covered. Many examples of ordinal sums of t-norms on non-linearly ordered bounded lattices are provided.