The problem of characterizing those locally convex spaces satisfying the Mackey convergence condition is still open. Recently in [4], a partial description was given using compatible webs. In this paper, those results are extended by using quasi-sequentially webbed spaces (see Definition 1). In particular, it is shown that strictly barrelled spaces satisfy the Mackey convergence condition and that they are properly contained in the set of quasi-sequentially webbed spaces. A related problem is that of characterizing those locally convex spaces satisfying the so-called fast convergence condition. A partial solution to this problem is obtained. Several examples are given.