I have successfully managed to negotiate higher than the supposed maximum.
I applied for a regular SE role and when I stated my expectations the recruiter said that this is over 10% more than the maximum. I stated that if I go any lower than that I will have no reason to switch companies. He offered some sort of a signing bonus (which was rather small) and the promise of quickly getting into a more senior role, which would put me in a higher pay range. I didn't budge.
They came back 3 days later and actually offered me the senior role, the pay I wanted and a signing bonus that was 4x what previously offered and I accepted.
I applied for a regular SE role and when I stated my expectations the recruiter said that this is over 10% more than the maximum. I stated that if I go any lower than that I will have no reason to switch companies. He offered some sort of a signing bonus (which was rather small) and the promise of quickly getting into a more senior role, which would put me in a higher pay range. I didn't budge.
They came back 3 days later and actually offered me the senior role, the pay I wanted and a signing bonus that was 4x what previously offered and I accepted.