Researchers use interference models based on exposure mappings to facilitate estimation of causal effects in randomized experiments with interference. To test the veracity of such models, researchers can use specification tests that aim to detect departures from the stipulated model. However, existing tests suffer from poor power and are often unable to detect important model violations. The main result in this paper is to show that the specification testing problem for exposure mapping models is inherently difficult, and the poor power of existing tests is inescapable. In particular, the worst-case Type I and Type II error rates must sum to one for any specification test of such models, ruling out the existence of a uniformly consistent test. This is the worst-case overall error rate achieved by a naive test that discards all data and arbitrarily rejects the null at random; the testing problem is in this sense impossible. This negative result holds true for all exposure mappings, all sample sizes, for uniformly bounded outcomes, and for alternatives that are maximally separated from the null. While some tests can detect some type of departures from the null model, there will always be relevant departures from the null that are undetectable. Informative specification tests must therefore restrict the alternative model against which they seek to attain power for, beyond the restrictions imposed by the exposure mappings alone. We illustrate this by providing a uniformly consistent test for differentiating no-interference from a network-linear-in-means model.