On some engines, not all engine parameters are limitation protected. On our RB211s, we had no protection on EGT. Other parameters were protected from exceedence. On one very heavy and hot take-off from Bombay, we had a power shortfal on a couple of engines. We had an allowed shortfall of power limit- .02EPR per engine or .06 across all 4 engines. It took some pretty rapid calculations by the Flight Enfgineer who could call Go/No Go. I drew attention to it, and being new on type, he pushed the thrust lever open to get rid of it. We had several red EGT lights on during the take-off. When airborne we looked at it and realised we'd exceeded max EGT by trying to chase N1/N2/N3 and EPR up to limits. It resulted in an engine change, 250 pax in hotels in Perth, and a special ferry flight to bring in a new engine (and a Qantas mechanic sliding off the top of the engine in a storm and breaking something).
That's not going into my memoirs!