Network models are used to study interconnected systems across many physical, biological, and social disciplines. Such models often assume a particular network-generating mechanism, which when fit to data produces estimates of mechanism-specific parameters that describe how systems function. For instance, a social network model might assume new individuals connect to others with probability proportional to their number of pre-existing connections ('preferential attachment'), and then estimate the disparity in interactions between famous and obscure individuals with similar qualifications. However, without a means of testing the relevance of the assumed mechanism, conclusions from such models could be misleading. Here we introduce a simple empirical approach which can mechanistically classify arbitrary network data. Our approach compares empirical networks to model networks from a user-provided candidate set of mechanisms, and classifies each network--with high accuracy--as originating from either one of the mechanisms or none of them. We tested 373 empirical networks against five of the most widely studied network mechanisms and found that most (228) were unlike any of these mechanisms. This raises the possibility that some empirical networks arise from mixtures of mechanisms. We show that mixtures are often unidentifiable because different mixtures can produce functionally equivalent networks. In such systems, which are governed by multiple mechanisms, our approach can still accurately predict out-of-sample functional properties.