Covalent organic framework (COF) membranes are of great promise for energy-efficient separations. Thick, polycrystalline COF films have been reported to separate dyes, salts, bacteria, and nanoparticles on the basis of size-selective transport through ordered pores. Here, we show that these materials function as adsorbents, not as size-sieving membranes. Binding isotherms of several dyes typical of the COF membrane literature to three COF powder samples illustrate that COFs are high-capacity adsorbents with affinities that span a range of 3 orders of magnitude, trends which map onto previously reported separation behavior. Computational results suggest that observed differences in adsorption can be correlated to variable entropic gains driving the adsorption process. Polycrystalline COF pellets show volume-dependent and flow-rate dependent "rejection" of dyes, consistent with an adsorption-based removal mechanism. Previous reports of thick, polycrystalline COF membranes used low flow rates and small dye volumes to probe rejection capabilities, where membrane and adsorbent behavior is not distinguishable. A mixed dye separation experiment in flow shows affinity-dependent performance. These results necessitate a careful reexamination of the COF membrane literature, as separations based on differential transport through 2D COF pores remain an important yet unrealized frontier.