it works differently in different markets. in fresno, they don't charge the late fee for instance. I know here, if your bill was due the first your account would have a cut-off date of the 2nd. if the bill was not paid by the second it would be shut off until payment was received. The account stay's hotlined until the end of the billing cycle. While the account is hotlined for non-payment, making payment will restore your service, and your normal billing will resume, meaning you will have another regular bill due on the 1st again. If the account goes a whole billing cycle (~30 days) in hotlined status for non-payment, the account will be closed and the esn released and can be activated again, with first months payment and activation charge.
edit: If you have problems paying on your due-date due to a fixed monthly income or something, try going to a full service store. Here they will allow you to change your billing cycle one time on the life of the account. It is extremely rare however, for them to extend the grace period. You are more likely to get a service credit, only if you have a legitimate reason tho(incarceration, broken equipment etc..)