No, Paul did not say we are promised that we'd always use the escape that God has provided for the sin you can withstand. He said that God will not allow a temptation that you can not resist and provides a way out of the temptation. Paul was not wrong. The problem is sometimes Christians don't avail themselves of the escape God has provided.
Reminds me of the fellow who was stuck on his rooftop in a flood. He was praying to God for help.
Soon a man in a rowboat came by and the fellow shouted to the man on the roof, “Jump in, I can save you.”
The stranded fellow shouted back, “No, it’s OK, I’m praying to God and he is going to save me.”
So the rowboat went on.
Then a motorboat came by. “The fellow in the motorboat shouted, “Jump in, I can save you.”
To this the stranded man said, “No thanks, I’m praying to God and he is going to save me. I have faith.”
So the motorboat went on.
Then a helicopter came by and the pilot shouted down, “Grab this rope and I will lift you to safety.”
To this the stranded man again replied, “No thanks, I’m praying to God and he is going to save me. I have faith.”
So the helicopter reluctantly flew away.
Soon the water rose above the rooftop and the man drowned. He went to Heaven. He finally got his chance to discuss this whole situation with God, at which point he exclaimed, “I had faith in you but you didn’t save me, you let me drown. I don’t understand why!”
To this God replied, “I sent you a rowboat and a motorboat and a helicopter, what more did you expect?”