Edges around sprites from spritesheet

I'm trying to make a pixel-art game, and have up until now used point-filtering for my spritesheets. This gave rise to horrible flickering whenever the camera moved around, and after reading around on this forum I found out that it would be better to scale up the textures and use bilinear/trilinear filtering instead. When I do this, however, pixels from the surrounding sprites seem to "bleed" into the edges of my sprites, so that the sprite that is supposed to look like this

now looks like this (notice the edge on the left side)

and the flickering is several orders of magnitude worse. Each individual sprite is of the exactly correct size. How can I prevent this from happening? I'm using 4.6, and mip-maps are off.