A seal system between a row of buckets supported on a machine rotor and a surrounding stationary casing or stator includes a tip shroud secured at radially outer tips of each of the buckets, the tip shroud formed with a radially-projecting rail. A cellular seal structure is supported in the stationary stator in radial opposition to the tip shroud and the rail. The seal structure has an annular array of individual cells formed to provide continuous, substantially horizontal flow passages devoid of any radial obstruction along substantially an entire axial length dimension of the cellular seal structure to prevent flow about the tip shroud from turning radially inwardly.