The present invention provides a process for producing a deep-nested embossed paper product comprising the steps of delivering one or more plies of paper to an embossing apparatus and embossing the one or more plies of the paper between two opposed embossing cylinders. The one or more plies of paper have a first direction and a second direction that is perpendicular to the first direction where both the first and second directions are in the plane of the paper and the one or more plies of paper have a stretch characteristic in the first direction that is higher than the stretch characteristic in the second direction. Each of the embossing cylinders having a plurality of protrusions, each of which have a height, where the embossing protrusions are disposed in an overall non-random pattern where the respective overall non-random patterns on the cylinders are coordinated to each other and the two embossing cylinders are aligned such that the respective coordinated overall non-random patterns of embossing protrusions nest together such that the protrusions engage each other to a depth of greater than about 1.016 mm. The overall non-random pattern of protrusions comprises a plurality of emboss regions where each of the emboss regions comprising a fraction of the total number of protrusions in the overall non-random pattern. All of the protrusions within an embossing region have about the same height and the pattern of protrusions within an emboss region creates a localized primary line of stress on the paper as the plies of paper are embossed where the line of stress has a component in the first direction and a component in the second direction. The height of the protrusions within an embossing region having a higher line of stress component in the first direction is greater than the height of the protrusions in an embossing region having a lower line of stress component in the first direction.