Mutations in the spike protein of SARS-CoV-2 variants can compromise the effectiveness of therapeutic antibodies. Most clinical-stage therapeutic antibodies target the spike receptor binding domain (RBD), but variants often have multiple mutations in several spike regions. To help predict antibody potency against emerging variants, we evaluated 25 clinical-stage therapeutic antibodies for neutralization activity against 60 pseudoviruses bearing spikes with single or multiple substitutions in several spike domains, including the full set of substitutions in B.1.1.7 (Alpha), B.1.351 (Beta), P.1 (Gamma), B.1.429 (Epsilon), B.1.526 (Iota), A.23.1 and R.1 variants. We found that 14 of 15 single antibodies were vulnerable to at least one RBD substitution, but most combination and polyclonal therapeutic antibodies remained potent. Key substitutions in variants with multiple spike substitutions predicted resistance, but the degree of resistance could be modified in unpredictable ways by other spike substitutions that may reside outside of the RBD. These findings highlight the importance of assessing antibody potency in the context of all substitutions in a variant and show that epistatic interactions in spike can modify virus susceptibility to therapeutic antibodies.