I have an intermittent problem with the blower on my 2006 Town and Country; and like any intermittent problem, it's driving me nuts.  

The blower sometimes does not come on when you turn on the switch, but may come on as you are driving.  It never stops on its own, but once you shut it off, it may or may not start again.

Before I tear into troubleshooting, are there any known or common issues that cause the problem (for example, I had an old Toyota Celica that ATE resistor blocks every few years)?

I have the same problem in my wife's Tahoe the blower motor will not start on its own but you can bump or jar it and it will start working. The problem is that the brushes in the motor are bad and the motor needs to be replaced.

on the trucks they have a "resistor" that plugs it right next to the blower
it also feeds power to the blower.
both parts can fail

even crazier sounding some chrysler products the a/c is wired inline with the ignition switch
ignition gets worn out and fails to make a good contact for the a/c to work
normally can be diagnosed by jiggling the key in the ignition and seeing if the a/c blower responds to that

super technical terms

Thanksfor the replies.  I'll try them out when I go to work on it.

I've replaced alot of blower motors, resistors, and ignition switches in those vans that had excessive amp draw and burned wires and melted connectors. Unplug the blower motor and resistor and see if there's any signs of heat or arcing, if so replace the motor and resistor and check the ignition switch also. I'm thinking there might be a TSB but can't remember. Pigtail connectors are available if you find them melted.