A multi-bridge hypergraph is an h-uniform linear hypergraph consisting of some linear paths having common extremities. In this paper it is proved that the multisets of path lengths of two chromatically equivalent multi-bridge hypergraphs are equal provided the multiplicities of path lengths are bounded above by 2 ^h-1 − 2. Also, it is shown that h-uniform linear cycles of length m are not chromatically unique for every m, h ≥ 3.