Let D be a masa in B(H) where H is a separable Hilbert space. We find real numbers η₀<η₁<η₂<...<η₆ so that for every bounded, normal D-bimodule map Φ on B(H), either ∥Φ∥>η₆ or ∥Φ∥=η_k for some k ∈ {0,1,2,3,4,5,6}. When D is totally atomic, these maps are the idempotent Schur multipliers and we characterise those with norm η_k for 0 ≦ k ≦ 6. We also show that the Schur idempotents which keep only the diagonal and superdiagonal of an n × n matrix, or of an n × (n+1) matrix, both have norm (2/(n+1))cot(π/(2(n+1))), and we consider the average norm of a random idempotent Schur multiplier as a function of dimension. Many of our arguments are framed in the combinatorial language of bipartite graphs.