We look for characterizations of those locally convex spaces that satisfy the strict Mackey convergence condition within the context of spaces with webs. We will say that a locally convex space has a boundedly compatible web if it has a web of absolutely convex sets whose members behave like zero neighborhoods in a metrizable locally convex space. It will be shown that these locally convex spaces satisfy the strict Mackey convergence condition. One consequence of this result will be a characterization of boundedly retractive inductive limits. We will also prove that if E is locally complete and webbed, then the strict Mackey convergence condition is equivalent to E having a boundedly compatible web.