If you have a tiling texture like this as a smaller section of a larger texture, you need to add wrapped padding pixels. The reason you need to do this is linear interpolation. Imagine a horizontal row of pixels in a seamless, tiled image: 5 6 7 6 When they wrap, the edge pixels look nice and wrap consistently: 5 6 7 6 5 6…
Long shot, but I know in photoshop when you scale an image that is in a layer anti aliasing can mess around with the border pixels, so make sure you scale the original tiling texture in a flattened document, then copy and paste it in the main file. Other than that might be snapping your UVs to the pixels.
Hmm, there is no such thing has parts of pixels, so no. Is this piece the one with your tile covering the entire uv shell? If it tiles perfectly in photoshop then it is your uv island that is causing the issue.