Antibody solutions deviate from the dynamical and rheological response expected for globular proteins, especially as the volume fraction is increased. Experimental evidence shows that antibodies can reversibly bind to each other via F-ab and F-c domains and form larger structures (clusters) of several,antibodies. Here, we present a microscopic equilibrium model to account for the distribution of cluster sizes. Antibody clusters are modeled as polymers that can grow via reversible bonds either between two F-ab domains or between F-ab and F-c domain. We propose that the dynamical and rheological behavior is determined by molecular entanglements of the clusters. This entanglement does not occur at low concentrations where antibody-antibody binding contributes to the viscosity by increasing the effective size of the particles. The model explains the observed shear-thinning behavior of antibody solutions.