Let Γ= (X, R) denote a bipartite distance-regular graph with diameter d≥ 4, and fix a vertex x of Γ. The Terwilliger algebra of Γ with respect to x is the subalgebra T ofMatX (C) generated by A, E*0,E*1,⋯ , E*d, where A is the adjacency matrix ofΓ , and where E*i denotes the projection onto the i th subconstituent ofΓ with respect to x. Let W denote an irreducible T -module. W is said to be thin whenever dim E*iW≤ 1 (0 ≤i≤d). The endpoint of W is min{ i |E*iW≠= 0}. It is known that a thin irreducibleT -module of endpoint 2 has dimension d− 3, d− 2, ord− 1. Γ is said to be 2-homogeneous whenever for all i(1 ≤i≤d− 1 ) and for all x, y,z∈X with ∂(x, y) = 2,∂ (x, z) =i, ∂(y,z ) =i, the number | Γ1(x) ∩Γ1(y) ∩Γi−1(z)| is independent ofx , y, z. Nomura has classified the 2-homogeneous bipartite distance-regular graphs. In this paper we study a slightly weaker condition. Γ is said to be almost 2-homogeneous whenever for all i(1 ≤i≤d− 2 ) and for all x, y,z∈X with ∂(x, y) = 2,∂ (x, z) =i, ∂(y,z ) =i, the number | Γ1(x) ∩Γ1(y) ∩Γi−1(z)| is independent ofx , y, z. We prove that the following are equivalent: (i)Γ is almost 2-homogeneous; (ii)Γ has, up to isomorphism, a unique irreducible T -module of endpoint 2 and this module is thin. Moreover, Γ is 2-homogeneous if and only if (i) and (ii) hold and the unique irreducible T -module of endpoint 2 has dimension d− 3.