Unless I'm doing something wrong, I think this is a bug.
Large PNGs imported as planes render strangely in both the viewport and final render (Eevee). I had this same issue in 2.80 and now using 2.81. Using Windows 10.
When seen face on (top image) things look good but when the angle to the plane is tighter (2nd image) you get distant PNGs which are technically behind objects appearing in front of them.
Most of these PNGs are 30,000 pixels wide, with an alpha mask.
I use the add-on import images as planes, and import with Emit setting (see shader node settings below). Blend mode: alpha blend.
I've uploaded a short video here. You can see the problems starting from 20 secs. Before then everything is fine.