Suppose, through inadvertence or otherwise, you have missed the deadline to apply for an extension or change of your non-immigrant status while physically present in the United States.  Must you leave the country?  Are there any circumstances under which the failure to timely apply for an extension or change of non-immigrant status may be excused?

Yes, but under very limited circumstances.  According to the law, the failure to timely file applications for an extension of non-immigrant status or a change of non-immigrant status in the U.S. may be excused by a USCIS  District Director if circumstances warrant.  8 C.F.R. §248.1(b) and §214.(1)(c)(4).  Here is the regulation:

8 CFR §248.1(b)

b)  Timely filing and maintenance of status.  A change of status may not be approved for an alien who failed to maintain the previously accorded status or whose status expired before the application or petition was filed, except that failure to file before the period of previously authorized status expired may be excused in the discretion of the Service, and without separate application, where it is demonstrated at the time of filing that: