We manage a large portfolio and file numerous evictions each month on the 15th. Once they are filed the tenant owes whatever we have spent on the attorney and court fees if they want to stay in the home and come current. In a typical month we only move out 10-20% of those people on whom we have filed. It never ceases to amaze me how resourceful people can to come up with past rent in addition to the $200 or $300 of additional costs - sometimes they just need that pressure to come around. I have one tenant that we filed on a dozen times over 6 years before he finally moved - yes, the house was a mess but it was a mess when we took it over so our client deferred the turnover costs for 5.5 more years.
Since I have known tenants that have had a child, two aunts, 4 grandmothers and 2 mothers die in an 18 month period (there is no shame in fibbing to your landlord) exceptions for specific events are really difficult because you don't really ever know what is truthful. Get them in court, hammer out a deal in court that gives you the right to move them out if they default and go from there.