You need to check the stability of your antibodies. Antibodies might be aggregated. Another important parameter is the conditions of your coupling solution, particularly pH and salt concentration, which often affects binding of biomolecules with their target ligands or receptors.

I do not think that you have a packing problem, something changed in the coupling procedure

Did you follow the instruction manual exactly for coupling antibodies to the activated affinity media? It's important that you stick to the recommendations for the immobilization procedure like the duration of incubation, absence of primary amines in the reaction buffer (e.g. Tris can't be used in it), correct pH, and so on.

You can test a small sample of your affinity matrix with immobilized antibodies by performing a small-scale pull-down assay with your purified FLAG-tagged protein (in case you have it available) and detect it using Western blotting with the same free anti-FLAG antibodies.

I agree with Ilya. Check in what solvent your antibody is diluted (if its tris or some other primary amine containing buffer, you must do a buffer exchange before coupling your antibody). You may also check if there is no stabilizing agent in your antibody solution, such as BSA. Finally, the functional groups for coupling are very reactive. You may wanna check if the resin is still active or if for some reason (storage, aging...) it´s not active anymore. 

good luck

Usually antibodies after purification are neutralised in Tris Buffer. And even frozen in. If you use them "like is" everything could be wrong. Ensure to change the buffer (dialysis or column buffer eschange)