Monoclonal antibodies against the SARS-CoV-2 spike protein, notably, those developed by Regeneron Pharmaceuticals and Eli Lilly and Company have proven to provide protection against severe COVID-19. The emergence of SARS-CoV-2 variants with heavily mutated spike proteins raises the concern that the therapy could become less effective if any of the mutations disrupt epitopes engaged by the antibodies. In this study, we tested monoclonal antibodies REGN10933 and REGN10987 that are used in combination, for their ability to neutralize SARS-CoV-2 variants B.1.1.7, B.1.351, mink cluster 5 and COH.20G/677H. We report that REGN10987 maintains most of its neutralization activity against viruses with B.1.1.7, B.1.351 and mink cluster 5 spike proteins but that REGN10933 has lost activity against B.1.351 and mink cluster 5. The failure of REGN10933 to neutralize B.1.351 is caused by the K417N and E484K mutations in the receptor binding domain; the failure to neutralize the mink cluster 5 spike protein is caused by the Y453F mutation. The REGN10933 and REGN10987 combination was 9.1-fold less potent on B.1.351 and 16.2-fold less potent on mink cluster 5, raising concerns of reduced efficacy in the treatment of patients infected with variant viruses. The results suggest that there is a need to develop additional monoclonal antibodies that are not affected by the current spike protein mutations.