Fault Sensitivity Analysis is an attack on cryptographic implementations that exploits dependencies between the sensitive data and the intensity of an injected fault. Masking, an established Side-Channel Analysis countermeasure, was originally believed to resist Fault Sensitivity Analysis, until Moradi et al. presented a successful attack on several masked AES ASIC cores by leveraging Fault Sensitivity Analysis. However, the attacked masked implementations are known to be vulnerable to power analysis through glitches occurring from non-ideal gates in CMOS. This means that glitch-resistant masking schemes specifically have not been assessed against Fault Sensitivity Analysis. In this work we give a response to this matter and show that implementations protected with these glitch-resistant masking schemes provide Fault Sensitivity Analysis resistance by design. We argue our claims through a theoretical elaboration and provide further evidence through simulations for both ASIC and FPGA platforms. In our setup we give the attackers numerous, often unrealistic, advantages, only to see the attacks fail against glitch-resistant masking schemes.