usually the Maximum late fee is 5% of the monthly rent.
The landlord cannot let the rent pile up and then sue you , usually the landlord can only go back at most 3 months, unless there is extenuating circumstances, like he is in a coma, or in a nursing home or has altzheimers. . Then his attorney or appointed represenative could try and collect all the back rent.
Others will quote the ny law but hopefully this is more of a common sense answer.