In Florida, if you accept anything other than what is specified in the lease, in the eyes of the court you are agreeing to modify the terms of the lease and it may not be considered binding when/if you take the tenant to court.
Tenants that play these games are aware that if they can get you to accept partial payment and take no action then they can tell the court that you agreed to let them pay in different terms. Why else would you accept anything other than what is clearly defined in the lease.
If someone is late and has been a good tenant, I will collect a check from them for the late fees, make a copy of the check so I have proof that I am following the lease to the letter and then I might 'forget' to deposit the late fee check. That way my lease is binding and if the tenant has always been good, I help them out.
However conducting yourself in any manner that does not follow the lease to the letter will only lead to bad things for the landlord.