This depends--I think collection laws vary state to state and possibly within counties or municipalities. When I was working in Ohio, we wanted every penny of rent and utilities by the 5th and didn't accept anything less. Late fees started on the sixth, increasing daily, and if you were late the late fee had to be included as well or we wouldn't accept payment. If we didn't have all monies owed by the 15th of the month, we began eviction proceedings (adding court fees to your total charge), and that moved along pretty fast--you could be evicted within six weeks of the first day of the month that you didn't pay. Ohio is pretty economically depressed and this was a harsh system for our residents, but the company owner was pretty happy when he saw how much money he was making based on late fees.:dry:
Now I'm in Maryland, and we collect rents within $100 of the amount due. Our county law says that residents have until the 10th to pay and residents usually have two months of non-payment before we start eviction proceedings. Late fees are a fixed percentage of the rent and we are very limited in the amount of court fees that we can pass on to residents once we file--again, based on county law. Even then, eviction proceedings are slow as molasses and we could have more than three months between getting a writ (?) to evict and actually getting a bailiff date.
Moral of the story is...check with a lawyer who is familiar with the local housing jurisdiction codes. In some places, partial payment is considered "payment" and you wouldn't be entitled to the remainder; in others, turning down a rent payment within a certain amount owed may be grounds for "rejecting" payment and you could get in trouble for that too.