Only one fan is spinning on my dual-fan GPU

Oops, sorry for the double-post. It was an accident.

I think that Gigabyte uses this fan feature, where one fan spins and the stops, the other starts etc. It is use to reduce dust accumulation if I remember right.
At load both should start spinnig.

I advise that tou read your Video Card manual.

Also I think there should be a switch on the Video Card that manualy turns both fans.
Somebody correct me if I'm wrong.

I have the coolermaster HAF 932 full tower case. There is a huge 200MM side panel fan attached to a wire. This means you cannot completely remove the side panel without disconnecting the wire.

The cards idle at a nice and cool 33 *C with only one fan spinning.

You've got to get an extension cable for that side panel fan of the HAF, I've got 2 of those cabinets and have not had problems removing the side panel to do anything without disconnecting the FAN wire from the Mobo, either you have it plugged into a header on the mobo that's way off....
Either way how did you connect it in the first place if you can't get it off???
All you need to do is remove the locking screws from the back and just slide the side panel onto the top of the case.... it usually works if you connected the fan cable to the wrong mobo header....
Anyways, if you are getting 33C then it's not much of an issue....

Is 60-65 *C under heavy load good temperatures?

Under heavy load those are good temps, but still is that temp with both fans on? or just one fan on?
What you can do if also use a bright flashlight and peek into the case thru side panel mesh to try and see if the fans (both) are working under full load or not.....

I'll try, but it's late. Maybe I'll do that tomorrow.

Ok will get back to you when you give us the next bit of info.....

First I can't see a problem in disconnecting a wire (it should be pluged in a 3 pin connector on the Mobo )

Second I have a magazine that I'm reading now and there is an advertisment for a Gigabyte GTS 450 (I know it's totaly diffrent, but stay with me). So here is what they say:
Wind Force on a GTS 450 OC

Dual Power Mode : Both Fans rusn for 10 minutes, the take turns in 30 seconds.

Smart Alternate Mode : Fans take turns running in 4 minutes.

If GPU temp exeeds 70 C Both Fans turn automaticly.

Again this refers to a GTS 450. Sorry if there are some mistakes, but I'm translating it, so..

There should be a little switch on your HD5870 Gigabyte and a LED near it, to switch between the two modes.

You can also go into CCC, overdrive page and view temp., % usage and fan speeds.
Click on manual fan speed and up it to see if both fans respond.