To tackle the global antibiotic resistance crisis, antibiotic resistance acquired either vertically by chromosomal mutations or horizontally through antibiotic resistance genes (ARGs) have been studied. Yet, little is known about the interactions between the two, which may impact the evolution of antibiotic resistance. Here, we develop a multiplexed barcoded approach to assess the fitness of 144 mutant-ARG combinations in Escherichia coli subjected to eight different antibiotics at 11 different concentrations. While most interactions are neutral, we identify significant interactions for 12% of the mutant-ARG combinations. The ability of most ARGs to confer high-level resistance at a low fitness cost shields the selective dynamics of mutants at low drug concentrations. Therefore, high-fitness mutants are often selected regardless of their resistance level. Finally, we identify strong negative epistasis between two unrelated resistance mechanisms: the tetA tetracycline resistance gene and loss-of-function nuo mutations involved in aminoglycoside tolerance. Our study highlights important constraints that may allow better prediction and control of antibiotic resistance evolution.