By Illinois State Statues, your payment must be U.S. postmarked by the due date in order to be processed as a timely payment. If you are receiving a balance due statement, then your payment was not U.S. postmarked (metered mail is not acceptable) on time and penalties have accrued. Placing your payment in the mailbox does not guarantee the envelope will be postmarked on that day. If you are going to wait until the last day to mail your payment, the only way to guarantee the official postmark is to go to the local post office and ask to have the envelope hand stamped. 

Yes, partial payments can be made by mail or at the Treasurer's Office, with the understanding that the remaining balances due after the due dates could be subject to penalty.