Not possible past a certain level. This rule was put in place specifically to stop teams with a ton of money (and not subject to the final 4 rule) from going out and buying up everyone with huge signing bonuses.
Which "rule" are you speaking of? I'd like to look it up. It's my understanding that the final 8 teams have preclusions on signing FA's not already on their team but the other 24 do not. A final 8 team that loses a FA/FA's is allowed to pursue the same number lost, at values equal to or less than the contracts the players that left signed.
While this rule may be in place I'm still unsure of the legroom provided by the uncapped year to increase signing bonuses without the usual penalty. I don't know for sure if there even is any really but there seems to be alot of talk leaning that way and I just want to know definitively. I'm trying to look it up now but can't find anything specific.