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In the present paper we investigate conditions for the non-existence of a smoothable limit linear series on a curve of compact type such that two smooth curves are bridged by a chain of two elliptic curves. Combining this work with results on the existence of a smoothable limit linear series on such a curve, we show relations among Brill–Noether loci of codimension at most two in the moduli space of complex curves. Specifically, Brill–Noether loci of codimension two have mutually distinct supports. [ABSTRACT FROM AUTHOR]