In 2010, representatives from the Computer History Museum visited a company in Texas still using an IBM 402 mainframe for everyday accounting jobs.
The photographs on for the CHM trip are pretty interesting, especially the punch card jams: http://ibm-1401.info/402.html
The article below mentions the CHM trip to Texas, and a few other old computers still in use:
If It Ain’t Broke, Don’t Fix It: Ancient Computers in Use Today, by Benj Edwards at PCWorld
The biggest problem with maintaining such ancient computer systems is that the original technicians who knew how to configure and maintain them have long since retired or passed away, so no one is left with the knowledge required to fix them if they break.