Foreclosure is a state law process. If under the law of your state the Demand Letter is the first notice, then it cannot be sent until the consumer is 120 days past due.
If the borrower makes a legal payment (in U.S. Dollars and meets any other requirement you may impose) the bank must accept the payment and credit the payment as of the date of receipt, unless other payment conditions have been established.
It is hard to image this scenario becoming common. Apparently the consumer has the ability to make a monthly payment, but chooses to remain delinquent thereby destroying his or her credit history and paying continuing late charges.