Cross-lingual word embeddings are an increasingly important reseource in cross-lingual methods for NLP, particularly for their role in transfer learning and unsupervised machine translation, purportedly opening up the opportunity for NLP applications for low-resource languages.  However, most research in this area implicitly expects the availablility of vast monolingual corpora for training embeddings, a scenario which is not realistic for many of the world's languages.  Moreover, much of the reporting of the performance of cross-lingual word embeddings is based on a fairly narrow set of mostly European language pairs.  Our study examines the performance of cross-lingual alignment across a more diverse set of language pairs; controls for the effect of the corpus size on which the monolingual embedding spaces are trained; and studies the impact of spectral graph properties of the embedding spsace on alignment.  Through our experiments on a more diverse set of language pairs, we find that performance in bilingual lexicon induction is generally poor in heterogeneous pairs, and that even using a gold or heuristically derived dictionary has little impact on the performance on these pairs of languages.  We also find that the performance for these languages only increases slowly with corpus size.  Finally, we find a moderate correlation between the isospectral difference of the source and target embeddings and the performance of bilingual lexicon induction.  We infer that methods other than cross-lingual alignment may be more appropriate in the case of both low resource languages and heterogeneous language pairs.