I have a 2006 Mazda 3 and usually leave the cabin fan in the on position (1, 2, 3, or 4). About two months ago, there was a 30 second delay between start of the engine and the cabin fan cutting on. This time gap became longer with each start until now it is in the range of 20 minutes. The gap exists if the AC is off or on. Recently, on an unusually humid day, the time gap dropped to about 5 minutes before the fan came on. It also seems to take a little less time on unusually hot days. Any suggestions?

 

If you change the 1-4 position does it come on at all?
You can try wiggling the fuse or pulling the dash apart to see if there's just a loose connector.
The blower motor has it's own relay, so you could try there first, see if it's loose.
Pull the dash apart and see if the switch has a bad connection.
Other than that it could be the resistor pack that controls the fan speed. If you're in a way humid area it could have corroded - though those usually last a decade before getting a disconnect.
I'd guess you get at the motor from underneath, not sure how much of the dash you need to pull apart. But the resistor pack is located at the motor connection.