Here's an interesting article from the Cooke website that discusses the resolution requirements of 4K:
According to that, 35 line pairs per mm is about as fine a detail as a 4K sensor can make out. Virtually every lens I've ever tested can easily resolve that, often it's more than 5 times that in the centre.
But of course Phil is right that resolution is a more complex thing than one measurement can describe - MTF charts for lenses will have different curves for varying contrast percentages and line pair detail, whether it's the centre or the corners of the frame, and whether the line pair is sagittal or meridional.
All very complicated sounding, but the point being, the performance of a lens is a multifaceted thing. Usually what lets a lens down is the aberration correction, how well things like colour fringing, astigmatism, coma and flare have been controlled. Often these aberrations are worse towards the edge of frame. On zooms, anamorphics and wide angles distortion can be an issue. Sometimes it can be the out of focus areas that people like to assess, or the way a sharp area rolls off into blur. The colour rendition between lenses can vary, very important in cinematography where sequential images need to match. The image can change magnification as you rack focus, or subtly shift. Very often it's the physical constraints that make a lens unsuitable - the poor markings, the minimum focus, the size, the mechanics.
Generally resolution itself is not the problem with low cost or vintage or stills lenses.