I don't think there's a trigger in the sense that it depends much on what you do. My hunch, and it's only that, is that FA's have a pre-determined threshold of either money or time. Some guys sign on the first possible cycle - even once in awhile a really good Type A. Other guys my sign once a certain $ threshold is reached. Still others (perhaps 40-50% of the best FA's) won't sign until the 2nd to last cycle.
I've offered max, led for 3-4 cycles, then got outbid (the winner had better coaches). Other times, I've held the lead for 5-6 cycles, lost the lead, gained it back, then the FA signed without giving me a chance to counter. Sometimes you get a "Sports Agency", other times you get "Boras". It's a pretty random process - and it's not easy. Since there are no Yankee and Red Sox franchises who get to play with an unfair advantage, it's tough to get the guys you want.