I had the same problem with my electric dryer a few years ago...it was a failed heating element. I've seen it on several other electric clothes dryers as well. The heating elements break over time due to repeated heating and cooling.
Sometimes it's hard to see the break with the naked eye because the heat is what makes it open, but it's easy to check the element using a continuity tester.
The reason the dryer keeps running is due to, as hardwareman said, the humidity sensor is operating as it should.
A new heating elements can usually be bought at a local appliance parts supply house as they are a common expendable item. They are relatively easy to change by removing the ducting that is screwed onto the back of the dryer. While you are in there you can clean the blower wheel and duct of all the accumulated lint and hair ties...ask me how I know!
The elements cost about $80.00 on average.