According to my experience,using textures with dimension more than 2048 width or height is not recommended.Devices like mobiles wont support more than this resolution and in PC also there are limitation to texture dimension based on the graphics card.
Things you should keep in mind to use textures for 2D games.
1)Try to use power of two sizes (512,1024,2048 pixels etc).
2)For tablet use texture with size 1024x1024 or less.
3)The trick of getting the true texture quality of the texture in game is to set TrueColor in the texture import settings.
3)Don't use MipMapping.
Hope this helps..!!!
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