Consoles while technically being able to support the screen resolution, will lack graphical horsepower to drive complex 3D games at 4k resolutions with a smooth frame rate.
Performance scales inversly to screen resolution, and 'll honestly be surprised if some of the really big titles (like BF4 for example) even run at full 1080p, I don't think the console GPUs will pack enough punch for that.
Much less 4x more power they'd need for 4k when compared to 1080p
I've driven a 2560x1600 panel for gaming for many years now and that's approx 2x HD and that takes some serious hardware to power, we're talking multiple high end graphics cards, way out of reach of the consoles I'm afraid.
Some 2D indy games might look nice in 4 though.