Most multi-fidelity schemes for optimization or reliability assessment rely on regression surrogates, such as Gaussian processes. Contrary to these approaches, we propose a classification-based multi-fidelity scheme for reliability assessment. This technique leverages multi-fidelity information to locally construct failure boundaries using support vector machine (SVM) classifiers. SVMs are subsequently used to estimate the probability of failure using Monte Carlo simulations. The use of classification has several advantages: It can handle discontinuous responses and reduce the number of function evaluations in the case of a large number of failure modes. In addition, in the context of multi-fidelity techniques, classification enables the identification of regions where the predictions (e.g., failure or safe) from the various fidelities are identical. At the core of the proposed scheme is an adaptive sampling routine driven by the probability of classification inconsistency between the models. This sampling routine explores sparsely sampled regions of inconsistency between the models of various fidelity to iteratively refine the approximation of the failure domain boundaries. A lookahead scheme, which looks one step into the future without any model evaluations, is used to selectively filter adaptive samples that do not induce substantial changes in the failure domain boundary approximation. The model management strategy is based on a framework that adaptively identifies a neighborhood of no confidence between the models. The proposed scheme is tested on analytical examples of dimensions ranging from 2 to 10, and finally applied to assess the reliability of a miniature shell and tube heat exchanger.