We present a bound on the number of sources identifiable in a class of array processing models with multiple parameters and signals per source. The bound is applied to determine the maximum number of uniquely resolvable plane-wave sources in various acoustic and electromagnetic vector-sensor models. We examine the use of a priori information about the sources, the effects of known and unknown noise characteristics, and the presence of nuisance parameters. Connections between identifiability and existence of the Cramer-Rao bound (CRB) are investigated. We show quantitatively how assumptions about the parameters can fundamentally affect the maximum number of identifiable sources.